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<p><a id="X823949B17B185DFA" name="X823949B17B185DFA"></a></p>
<div class="ChapSects"><a href="chap4.html#X823949B17B185DFA">4 <span class="Heading">New Objects and Utility Functions Provided by the
<strong class="pkg">AtlasRep</strong> Package</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X8121E9567A7137C9">4.1 <span class="Heading">Straight Line Decisions</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X8787E2EC7DB85A89">4.1-1 IsStraightLineDecision</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82AFAD9F7FA5CE8A">4.1-2 LinesOfStraightLineDecision</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7B1A43427BD97FDF">4.1-3 NrInputsOfStraightLineDecision</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X82A3632782E45F35">4.1-4 ScanStraightLineDecision</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X825C4E4180F3D989">4.1-5 StraightLineDecision</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E7B328A84685480">4.1-6 ResultOfStraightLineDecision</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C94ECAC8583CEAE">4.1-7 <span class="Heading">Semi-Presentations and Presentations</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7C13D08C7D55E20A">4.1-8 AsStraightLineDecision</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EA613C57DDC67D5">4.1-9 StraightLineProgramFromStraightLineDecision</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7BE856BC785A9E8F">4.2 <span class="Heading">Black Box Programs</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X87CAF2DE870D0E3B">4.2-1 IsBBoxProgram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7EA20532868F9863">4.2-2 ScanBBoxProgram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D211A5D8602B330">4.2-3 RunBBoxProgram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X869BACFB80A3CC87">4.2-4 ResultOfBBoxProgram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X826ACFE887E0B6B8">4.2-5 AsBBoxProgram</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7D36DFA87C8B2C48">4.2-6 AsStraightLineProgram</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X87E1F08D80C9E069">4.3 <span class="Heading">Representations of Minimal Degree</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7DC66D8282B2BB7F">4.3-1 MinimalRepresentationInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7E1B76DC86A8C405">4.3-2 MinimalRepresentationInfoData</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X79C4C9F683E919C9">4.3-3 SetMinimalRepresentationInfo</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7FC33DFF8481F8D1">4.3-4 <span class="Heading">Criteria Used to Compute Minimality Information</span></a>
</span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X85217A9578DA034A">4.3-5 AGR_TestMinimalDegrees</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X7F31A7CB841FE63F">4.3-6 BrowseMinimalDegrees</a></span>
</div>
<div class="ContSect"><span class="nocss"> </span><a href="chap4.html#X7DD6CE0E79AAD61B">4.4 <span class="Heading">Bibliographies of Sporadic Simple Groups</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap4.html#X84ED4FC182C28198">4.4-1 BrowseBibliographySporadicSimple</a></span>
</div>
</div>
<h3>4 <span class="Heading">New Objects and Utility Functions Provided by the
<strong class="pkg">AtlasRep</strong> Package</span></h3>
<p>This chapter describes <strong class="pkg">GAP</strong> objects and functions that are provided in the <strong class="pkg">AtlasRep</strong> package but that might be of general interest.</p>
<p>The new objects are straight line decisions (see Section <a href="chap4.html#X8121E9567A7137C9"><b>4.1</b></a>) and black box programs (see Section <a href="chap4.html#X7BE856BC785A9E8F"><b>4.2</b></a>).</p>
<p>The new functions are concerned with representations of minimal degree, see Section <a href="chap4.html#X87E1F08D80C9E069"><b>4.3</b></a>.</p>
<p><a id="X8121E9567A7137C9" name="X8121E9567A7137C9"></a></p>
<h4>4.1 <span class="Heading">Straight Line Decisions</span></h4>
<p><em>Straight line decisions</em> are similar to straight line programs (see Section <a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: Straight Line Programs</b></a>) but return <code class="keyw">true</code> or <code class="keyw">false</code>. A straight line decisions checks a property for its inputs. An important example is to check whether a given list of group generators is in fact a list of standard generators (cf. Section<a href="chap2.html#X795DB7E486E0817D"><b>2.3</b></a>) for this group.</p>
<p>A straight line decision in <strong class="pkg">GAP</strong> is represented by an object in the category <code class="func">IsStraightLineDecision</code> (<a href="chap4.html#X8787E2EC7DB85A89"><b>4.1-1</b></a>) that stores a list of "lines" each of which has one of the following three forms.</p>
<ol>
<li><p>a nonempty dense list l of integers,</p>
</li>
<li><p>a pair [ l, i ] where l is a list of form 1. and i is a positive integer,</p>
</li>
<li><p>a list [<code class="code">"Order"</code>, i, n ] where i and n are positive integers.</p>
</li>
</ol>
<p>The first two forms have the same meaning as for straight line programs (see Section <a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: Straight Line Programs</b></a>), the last form means a check whether the element stored at the label i-th has the order n.</p>
<p>For the meaning of the list of lines, see <code class="func">ResultOfStraightLineDecision</code> (<a href="chap4.html#X7E7B328A84685480"><b>4.1-6</b></a>).</p>
<p>Straight line decisions can be constructed using <code class="func">StraightLineDecision</code> (<a href="chap4.html#X825C4E4180F3D989"><b>4.1-5</b></a>), defining attributes for straight line decisions are <code class="func">NrInputsOfStraightLineDecision</code> (<a href="chap4.html#X7B1A43427BD97FDF"><b>4.1-3</b></a>) and <code class="func">LinesOfStraightLineDecision</code> (<a href="chap4.html#X82AFAD9F7FA5CE8A"><b>4.1-2</b></a>), an operation for straight line decisions is <code class="func">ResultOfStraightLineDecision</code> (<a href="chap4.html#X7E7B328A84685480"><b>4.1-6</b></a>).</p>
<p>Special methods applicable to straight line decisions are installed for the operations <code class="func">Display</code> (<a href="../../../doc/htm/ref/CHAP006.htm#SECT003"><b>Reference: Display</b></a>), <code class="func">IsInternallyConsistent</code> (<a href="../../../doc/htm/ref/CHAP012.htm#SECT008"><b>Reference: IsInternallyConsistent</b></a>), <code class="func">PrintObj</code> (<a href="../../../doc/htm/ref/CHAP006.htm#SECT003"><b>Reference: PrintObj</b></a>), and <code class="func">ViewObj</code> (<a href="../../../doc/htm/ref/CHAP006.htm#SECT003"><b>Reference: ViewObj</b></a>).</p>
<p>For a straight line decision <var class="Arg">prog</var>, the default <code class="func">Display</code> (<a href="../../../doc/htm/ref/CHAP006.htm#SECT003"><b>Reference: Display</b></a>) method prints the interpretation of <var class="Arg">prog</var> as a sequence of assignments of associative words and of order checks; a record with components <code class="code">gensnames</code> (with value a list of strings) and <code class="code">listname</code> (a string) may be entered as second argument of <code class="func">Display</code> (<a href="../../../doc/htm/ref/CHAP006.htm#SECT003"><b>Reference: Display</b></a>), in this case these names are used, the default for <code class="code">gensnames</code> is <code class="code">[ g1, g2, </code>...<code class="code"> ]</code>, the default for <var class="Arg">listname</var> is r.</p>
<p><a id="X8787E2EC7DB85A89" name="X8787E2EC7DB85A89"></a></p>
<h5>4.1-1 IsStraightLineDecision</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsStraightLineDecision</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Each straight line decision in <strong class="pkg">GAP</strong> lies in the category <code class="func">IsStraightLineDecision</code>.</p>
<p><a id="X82AFAD9F7FA5CE8A" name="X82AFAD9F7FA5CE8A"></a></p>
<h5>4.1-2 LinesOfStraightLineDecision</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> LinesOfStraightLineDecision</code>( <var class="Arg">prog</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>the list of lines that define the straight line decision.</p>
<p>This defining attribute for the straight line decision <var class="Arg">prog</var> (see <code class="func">IsStraightLineDecision</code> (<a href="chap4.html#X8787E2EC7DB85A89"><b>4.1-1</b></a>)) corresponds to <code class="func">LinesOfStraightLineProgram</code> (<a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: LinesOfStraightLineProgram</b></a>) for straight line programs.</p>
<table class="example">
<tr><td><pre>
gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ],
> [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] );
<straight line decision>
gap> LinesOfStraightLineDecision( dec );
[ [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 1, 2 ], [ "Order", 2, 3 ],
[ "Order", 3, 5 ] ]
</pre></td></tr></table>
<p><a id="X7B1A43427BD97FDF" name="X7B1A43427BD97FDF"></a></p>
<h5>4.1-3 NrInputsOfStraightLineDecision</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> NrInputsOfStraightLineDecision</code>( <var class="Arg">prog</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>the number of inputs required for the straight line decision.</p>
<p>This defining attribute corresponds to <code class="func">NrInputsOfStraightLineProgram</code> (<a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: NrInputsOfStraightLineProgram</b></a>).</p>
<table class="example">
<tr><td><pre>
gap> NrInputsOfStraightLineDecision( dec );
2
</pre></td></tr></table>
<p><a id="X82A3632782E45F35" name="X82A3632782E45F35"></a></p>
<h5>4.1-4 ScanStraightLineDecision</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ScanStraightLineDecision</code>( <var class="Arg">string</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a record containing the straight line decision, or <code class="keyw">fail</code>.</p>
<p>Let <var class="Arg">string</var> be a string that encodes a straight line decision in the sense that it consists of the lines listed for <code class="func">ScanStraightLineProgram</code> (<a href="chap5.html#X7D6617E47B013A37"><b>5.4-1</b></a>), except that <code class="code">oup</code> lines are not allowed, and instead lines of the following form may occur.</p>
<dl>
<dt><strong class="Mark"><code class="code">chor <var class="Arg">a</var> <var class="Arg">b</var></code></strong></dt>
<dd><p>means that it is checked whether the order of the element at label <var class="Arg">a</var> is <var class="Arg">b</var>.</p>
</dd>
</dl>
<p><code class="func">ScanStraightLineDecision</code> returns a record containing as the value of its component <code class="code">program</code> the corresponding <strong class="pkg">GAP</strong> straight line decision (see <code class="func">IsStraightLineDecision</code> (<a href="chap4.html#X8787E2EC7DB85A89"><b>4.1-1</b></a>)) if the input string satisfies the syntax rules stated above, and returns <code class="keyw">fail</code> otherwise. In the latter case, information about the first corrupted line of the program is printed if the info level of <code class="func">InfoCMeatAxe</code> (<a href="chap5.html#X78601C3A87921E08"><b>5.1-2</b></a>) is at least 1.</p>
<table class="example">
<tr><td><pre>
gap> str:= "inp 2\nchor 1 2\nchor 2 3\nmu 1 2 3\nchor 3 5";;
gap> prg:= ScanStraightLineDecision( str );
rec( program := <straight line decision> )
gap> prg:= prg.program;;
gap> Display( prg );
# input:
r:= [ g1, g2 ];
# program:
if Order( r[1] ) <> 2 then return false; fi;
if Order( r[2] ) <> 3 then return false; fi;
r[3]:= r[1]*r[2];
if Order( r[3] ) <> 5 then return false; fi;
# return value:
true
</pre></td></tr></table>
<p><a id="X825C4E4180F3D989" name="X825C4E4180F3D989"></a></p>
<h5>4.1-5 StraightLineDecision</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> StraightLineDecision</code>( <var class="Arg">lines[, nrgens]</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> StraightLineDecisionNC</code>( <var class="Arg">lines[, nrgens]</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>the straight line decision given by the list of lines.</p>
<p>Let <var class="Arg">lines</var> be a list of lists that defines a unique straight line decision (see <code class="func">IsStraightLineDecision</code> (<a href="chap4.html#X8787E2EC7DB85A89"><b>4.1-1</b></a>)); in this case <code class="func">StraightLineDecision</code> returns this program, otherwise an error is signalled. The optional argument <var class="Arg">nrgens</var> specifies the number of input generators of the program; if a list of integers (a line of form 1. in the definition above) occurs in <var class="Arg">lines</var> then this number is not determined by <var class="Arg">lines</var> and therefore <em>must</em> be specified by the argument <var class="Arg">nrgens</var>; if not then <code class="func">StraightLineDecision</code> returns <code class="keyw">fail</code>.</p>
<p><code class="func">StraightLineDecisionNC</code> does the same as <code class="func">StraightLineDecision</code>, except that the internal consistency of the program is not checked.</p>
<p><a id="X7E7B328A84685480" name="X7E7B328A84685480"></a></p>
<h5>4.1-6 ResultOfStraightLineDecision</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ResultOfStraightLineDecision</code>( <var class="Arg">prog, gens[, orderfunc]</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b><code class="keyw">true</code> if all checks succeed, otherwise <code class="keyw">false</code>.</p>
<p><code class="func">ResultOfStraightLineDecision</code> evaluates the straight line decision (see <code class="func">IsStraightLineDecision</code> (<a href="chap4.html#X8787E2EC7DB85A89"><b>4.1-1</b></a>)) <var class="Arg">prog</var> at the group elements in the list <var class="Arg">gens</var>.</p>
<p>The function for computing the order of a group element can be given as the optional argument <var class="Arg">orderfunc</var>. For example, this may be a function that gives up at a certain limit if one has to be aware of extremely huge orders in failure cases.</p>
<p>The <em>result</em> of a straight line decision with lines p_1, p_2, ..., p_k when applied to <var class="Arg">gens</var> is defined as follows.</p>
<dl>
<dt><strong class="Mark">(a)</strong></dt>
<dd><p>First a list r of intermediate values is initialized with a shallow copy of <var class="Arg">gens</var>.</p>
</dd>
<dt><strong class="Mark">(b)</strong></dt>
<dd><p>For i <= k, before the i-th step, let r be of length n. If p_i is the external representation of an associative word in the first n generators then the image of this word under the homomorphism that is given by mapping r to these first n generators is added to r. If p_i is a pair [ l, j ], for a list l, then the same element is computed, but instead of being added to r, it replaces the j-th entry of r. If p_i is a triple [<code class="code">"Order"</code>, i, n ] then it is checked whether the order of r[i] is n; if not then <code class="keyw">false</code> is returned immediately.</p>
</dd>
<dt><strong class="Mark">(c)</strong></dt>
<dd><p>If all k lines have been processed and no order check has failed then <code class="keyw">true</code> is returned.</p>
</dd>
</dl>
<p>Here are some examples.</p>
<table class="example">
<tr><td><pre>
gap> dec:= StraightLineDecision( [ ], 1 );
<straight line decision>
gap> ResultOfStraightLineDecision( dec, [ () ] );
true
</pre></td></tr></table>
<p>The above straight line decision <code class="code">dec</code> returns <code class="keyw">true</code> –for <em>any</em> input of the right length.</p>
<table class="example">
<tr><td><pre>
gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ],
> [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] );
<straight line decision>
gap> LinesOfStraightLineDecision( dec );
[ [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 1, 2 ], [ "Order", 2, 3 ],
[ "Order", 3, 5 ] ]
gap> ResultOfStraightLineDecision( dec, [ (), () ] );
false
gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,4,5) ] );
true
</pre></td></tr></table>
<p>The above straight line decision admits two inputs; it tests whether the orders of the inputs are 2 and 3, and the order of their product is 5.</p>
<p><a id="X7C94ECAC8583CEAE" name="X7C94ECAC8583CEAE"></a></p>
<h5>4.1-7 <span class="Heading">Semi-Presentations and Presentations</span></h5>
<p>We can associate a <em>finitely presented group</em> F / R to each straight line decision <var class="Arg">dec</var>, say, as follows. The free generators of the free group F are in bijection with the inputs, and the defining relators generating R as a normal subgroup of F are given by those words w^k for which <var class="Arg">dec</var> contains a check whether the order of w equals k.</p>
<p>So if <var class="Arg">dec</var> returns <code class="keyw">true</code> for the input list [ g_1, g_2, ..., g_n ] then mapping the free generators of F to the inputs defines an epimorphism Phi from F to the group G, say, that is generated by these inputs, such that R is contained in the kernel of Phi.</p>
<p>(Note that "satisfying <var class="Arg">dec</var>" is a stronger property than "satisfying a presentation". For example, < x | x^2 = x^3 = 1 > is a presentation for the trivial group, but the straight line decision that checks whether the order of x is both 2 and 3 clearly always returns <code class="keyw">false</code>.)</p>
<p>The <strong class="pkg">ATLAS</strong> of Group Representations contains the following two kinds of straight line decisions.</p>
<ul>
<li><p>A <em>presentation</em> is a straight line decision <var class="Arg">dec</var> that is defined for a set of standard generators of a group G and that returns <code class="keyw">true</code> if and only if the list of inputs is in fact a sequence of such standard generators for G. In other words, the relators derived from the order checks in the way described above are defining relators for G, and moreover these relators are words in terms of standard generators. (In particular the kernel of the map Phi equals R whenever <var class="Arg">dec</var> returns <code class="keyw">true</code>.)</p>
</li>
<li><p>A <em>semi-presentation</em> is a straight line decision <var class="Arg">dec</var> that is defined for a set of standard generators of a group G and that returns <code class="keyw">true</code> for a list of inputs <em>that is known to generate a group isomorphic with G</em> if and only if these inputs form in fact a sequence of standard generators for G. In other words, the relators derived from the order checks in the way described above are <em>not necessarily defining relators</em> for G, but if we assume that the g_i generate G then they are standard generators. (In particular, F / R may be a larger group than G but in this case Phi maps the free generators of F to standard generators of G.)</p>
<p>More about semi-presentations can be found in <a href="chapBib.html#biBNW05">[NW05]</a>.</p>
</li>
</ul>
<p>Available presentations and semi-presentations are listed by <code class="func">DisplayAtlasInfo</code> (<a href="chap2.html#X79DACFFA7E2D1A99"><b>2.5-1</b></a>), they can be accessed via <code class="func">AtlasProgram</code> (<a href="chap2.html#X801F2E657C8A79ED"><b>2.5-3</b></a>). (Clearly each presentation is also a semi-presentation. So a semi-presentation for some standard generators of a group is regarded as available whenever a presentation for these standard generators and this group is available.)</p>
<p>Note that different groups can have the same semi-presentation. We illustrate this with an example that is mentioned in <a href="chapBib.html#biBNW05">[NW05]</a>. The groups L_2(7) cong L_3(2) and L_2(8) are generated by elements of the orders 2 and 3 such that their product has order 7, and no further conditions are necessary to define standard generators.</p>
<table class="example">
<tr><td><pre>
gap> check:= AtlasProgram( "L2(8)", "check" );
rec( program := <straight line decision>, standardization := 1,
identifier := [ "L2(8)", "L28G1-check1", 1, 1 ], groupname := "L2(8)" )
gap> gens:= AtlasGenerators( "L2(8)", 1 );
rec( generators := [ (1,2)(3,4)(6,7)(8,9), (1,3,2)(4,5,6)(7,8,9) ],
groupname := "L2(8)", standardization := 1, repnr := 1,
identifier := [ "L2(8)", [ "L28G1-p9B0.m1", "L28G1-p9B0.m2" ], 1, 9 ],
p := 9, id := "", size := 504 )
gap> ResultOfStraightLineDecision( check.program, gens.generators );
true
gap> gens:= AtlasGenerators( "L3(2)", 1 );
rec( generators := [ (2,4)(3,5), (1,2,3)(5,6,7) ], groupname := "L3(2)",
standardization := 1, repnr := 1,
identifier := [ "L3(2)", [ "L27G1-p7aB0.m1", "L27G1-p7aB0.m2" ], 1, 7 ],
p := 7, id := "a", size := 168 )
gap> ResultOfStraightLineDecision( check.program, gens.generators );
true
</pre></td></tr></table>
<p><a id="X7C13D08C7D55E20A" name="X7C13D08C7D55E20A"></a></p>
<h5>4.1-8 AsStraightLineDecision</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AsStraightLineDecision</code>( <var class="Arg">bbox</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>an equivalent straight line decision for the given black box program, or <code class="keyw">fail</code>.</p>
<p>For a black box program (see <code class="func">IsBBoxProgram</code> (<a href="chap4.html#X87CAF2DE870D0E3B"><b>4.2-1</b></a>)) <var class="Arg">bbox</var>, <code class="func">AsStraightLineDecision</code> returns a straight line decision (see <code class="func">IsStraightLineDecision</code> (<a href="chap4.html#X8787E2EC7DB85A89"><b>4.1-1</b></a>)) with the same output as <var class="Arg">bbox</var>, in the sense of <code class="func">AsBBoxProgram</code> (<a href="chap4.html#X826ACFE887E0B6B8"><b>4.2-5</b></a>), if such a straight line decision exists, and <code class="keyw">fail</code> otherwise.</p>
<table class="example">
<tr><td><pre>
gap> lines:= [ [ "Order", 1, 2 ], [ "Order", 2, 3 ],
> [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ];;
gap> dec:= StraightLineDecision( lines, 2 );
<straight line decision>
gap> bboxdec:= AsBBoxProgram( dec );
<black box program>
gap> asdec:= AsStraightLineDecision( bboxdec );
<straight line decision>
gap> LinesOfStraightLineDecision( asdec );
[ [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ [ 1, 1, 2, 1 ], 3 ],
[ "Order", 3, 5 ] ]
</pre></td></tr></table>
<p><a id="X7EA613C57DDC67D5" name="X7EA613C57DDC67D5"></a></p>
<h5>4.1-9 StraightLineProgramFromStraightLineDecision</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> StraightLineProgramFromStraightLineDecision</code>( <var class="Arg">dec</var> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><b>Returns: </b>the straight line program associated to the given straight line decision.</p>
<p>For a straight line decision <var class="Arg">dec</var> (see <code class="func">IsStraightLineDecision</code> (<a href="chap4.html#X8787E2EC7DB85A89"><b>4.1-1</b></a>), <code class="func">StraightLineProgramFromStraightLineDecision</code> returns the straight line program (see <code class="func">IsStraightLineProgram</code> (<a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: IsStraightLineProgram</b></a>) obtained by replacing each line of type 3. (i.e, each order check) by an assignment of the power in question to a new slot, and by declaring the list of these elements as the return value.</p>
<p>This means that the return value describes exactly the defining relators of the presentation that is associated to the straight line decision, see <a href="chap4.html#X7C94ECAC8583CEAE"><b>4.1-7</b></a>.</p>
<p>For example, one can use the return value for printing the relators with <code class="func">StringOfResultOfStraightLineProgram</code> (<a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: StringOfResultOfStraightLineProgram</b></a>), or for explicitly constructing the relators as words in terms of free generators, by applying <code class="func">ResultOfStraightLineProgram</code> (<a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: ResultOfStraightLineProgram</b></a>) to the program and to these generators.</p>
<table class="example">
<tr><td><pre>
gap> dec:= StraightLineDecision( [ [ [ 1, 1, 2, 1 ], 3 ],
> [ "Order", 1, 2 ], [ "Order", 2, 3 ], [ "Order", 3, 5 ] ] );
<straight line decision>
gap> prog:= StraightLineProgramFromStraightLineDecision( dec );
<straight line program>
gap> Display( prog );
# input:
r:= [ g1, g2 ];
# program:
r[3]:= r[1]*r[2];
r[4]:= r[1]^2;
r[5]:= r[2]^3;
r[6]:= r[3]^5;
# return values:
[ r[4], r[5], r[6] ]
gap> StringOfResultOfStraightLineProgram( prog, [ "a", "b" ] );
"[ a^2, b^3, (ab)^5 ]"
gap> gens:= GeneratorsOfGroup( FreeGroup( "a", "b" ) );
[ a, b ]
gap> ResultOfStraightLineProgram( prog, gens );
[ a^2, b^3, a*b*a*b*a*b*a*b*a*b ]
</pre></td></tr></table>
<p><a id="X7BE856BC785A9E8F" name="X7BE856BC785A9E8F"></a></p>
<h4>4.2 <span class="Heading">Black Box Programs</span></h4>
<p><em>Black box programs</em> formalize the idea that one takes some group elements, forms arithmetic expressions in terms of them, tests properties of these expressions, executes conditional statements (including jumps inside the program) depending on the results of these tests, and eventually returns some result.</p>
<p>A specification of the language can be found in <a href="chapBib.html#biBNic06">[Nic06]</a>, see also</p>
<p><span class="URL"><a href="http://brauer.maths.qmul.ac.uk/Atlas/info/blackbox.html">http://brauer.maths.qmul.ac.uk/Atlas/info/blackbox.html</a></span>.</p>
<p>The <em>inputs</em> of a black box program may be explicit group elements, and the program may also ask for random elements from a given group. The <em>program steps</em> form products, inverses, conjugates, commutators, etc. of known elements, <em>tests</em> concern essentially the orders of elements, and the <em>result</em> is a list of group elements or <code class="keyw">true</code> or <code class="keyw">false</code> or <code class="keyw">fail</code>.</p>
<p>Examples that can be modeled by black box programs are</p>
<dl>
<dt><strong class="Mark"><em>straight line programs</em>,</strong></dt>
<dd><p>which require a fixed number of input elements and form arithmetic expressions of elements but do not use random elements, tests, conditional statements and jumps; the return value is always a list of elements; these programs are described in Section <a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: Straight Line Programs</b></a>.</p>
</dd>
<dt><strong class="Mark"><em>straight line decisions</em>,</strong></dt>
<dd><p>which differ from straight line programs only in the sense that also order tests are admissible, and that the return value is <code class="keyw">true</code> if all these tests are satisfied, and <code class="keyw">false</code> as soon as the first such test fails; they are described in Section <a href="chap4.html#X8121E9567A7137C9"><b>4.1</b></a>.</p>
</dd>
<dt><strong class="Mark"><em>scripts for finding standard generators</em>,</strong></dt>
<dd><p>which take a group and a function to generate a random element in this group but no explicit input elements, admit all control structures, and return either a list of standard generators or <code class="keyw">fail</code>; see <code class="func">ResultOfBBoxProgram</code> (<a href="chap4.html#X869BACFB80A3CC87"><b>4.2-4</b></a>) for examples.</p>
</dd>
</dl>
<p>In the case of general black box programs, currently <strong class="pkg">GAP</strong> provides only the possibility to read an existing program via <code class="func">ScanBBoxProgram</code> (<a href="chap4.html#X7EA20532868F9863"><b>4.2-2</b></a>), and to run the program using <code class="func">RunBBoxProgram</code> (<a href="chap4.html#X7D211A5D8602B330"><b>4.2-3</b></a>). The aim is not to write such programs in <strong class="pkg">GAP</strong>.</p>
<p>The special case of the "find" scripts mentioned above is also admissible as an argument of <code class="func">ResultOfBBoxProgram</code> (<a href="chap4.html#X869BACFB80A3CC87"><b>4.2-4</b></a>), which returns either the set of generators or <code class="keyw">fail</code>.</p>
<p>Contrary to the general situation, more support is provided for straight line programs and straight line decisions in <strong class="pkg">GAP</strong>, see Section <a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: Straight Line Programs</b></a> for functions that manipulate them (compose, restrict etc.).</p>
<p>The functions <code class="func">AsStraightLineProgram</code> (<a href="chap4.html#X7D36DFA87C8B2C48"><b>4.2-6</b></a>) and <code class="func">AsStraightLineDecision</code> (<a href="chap4.html#X7C13D08C7D55E20A"><b>4.1-8</b></a>) can be used to transform a general black box program object into a straight line program or a straight line decision if this is possible.</p>
<p>Conversely, one can create an equivalent general black box program from a straight line program or from a straight line decision with <code class="func">AsBBoxProgram</code> (<a href="chap4.html#X826ACFE887E0B6B8"><b>4.2-5</b></a>).</p>
<p>(Computing a straight line program related to a given straight line decision is supported in the sense of <code class="func">StraightLineProgramFromStraightLineDecision</code> (<a href="chap4.html#X7EA613C57DDC67D5"><b>4.1-9</b></a>).)</p>
<p>Note that none of these three kinds of objects is a special case of another: Running a black box program with <code class="func">RunBBoxProgram</code> (<a href="chap4.html#X7D211A5D8602B330"><b>4.2-3</b></a>) yields a record, running a straight line program with <code class="func">ResultOfStraightLineProgram</code> (<a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: ResultOfStraightLineProgram</b></a>) yields a list of elements, and running a straight line decision with <code class="func">ResultOfStraightLineDecision</code> (<a href="chap4.html#X7E7B328A84685480"><b>4.1-6</b></a>) yields <code class="keyw">true</code> or <code class="keyw">false</code>.</p>
<p><a id="X87CAF2DE870D0E3B" name="X87CAF2DE870D0E3B"></a></p>
<h5>4.2-1 IsBBoxProgram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsBBoxProgram</code>( <var class="Arg">obj</var> )</td><td class="tdright">( category )</td></tr></table></div>
<p>Each black box program in <strong class="pkg">GAP</strong> lies in the category <code class="func">IsBBoxProgram</code>.</p>
<p><a id="X7EA20532868F9863" name="X7EA20532868F9863"></a></p>
<h5>4.2-2 ScanBBoxProgram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ScanBBoxProgram</code>( <var class="Arg">string</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a record containing the black box program encoded by the input string, or <code class="keyw">fail</code>.</p>
<p>For a string <var class="Arg">string</var> that describes a black box program, e.g., the return value of <code class="func">StringFile</code> (<a href="../../../pkg/gapdoc/doc/chap6.html#X7E14D32181FBC3C3"><b>GAPDoc: StringFile</b></a>), <code class="func">ScanBBoxProgram</code> computes this black box program. If this is successful then the return value is a record containing as the value of its component <code class="code">program</code> the corresponding <strong class="pkg">GAP</strong> object that represents the program, otherwise <code class="keyw">fail</code> is returned.</p>
<p>As the first example, we construct a black box program that tries to find standard generators for the alternating group A_5; these standard generators are any pair of elements of the orders 2 and 3, respectively, such that their product has order 5.</p>
<table class="example">
<tr><td><pre>
gap> findstr:= "\
> set V 0\n\
> lbl START1\n\
> rand 1\n\
> ord 1 A\n\
> incr V\n\
> if V gt 100 then timeout\n\
> if A notin 1 2 3 5 then fail\n\
> if A noteq 2 then jmp START1\n\
> lbl START2\n\
> rand 2\n\
> ord 2 B\n\
> incr V\n\
> if V gt 100 then timeout\n\
> if B notin 1 2 3 5 then fail\n\
> if B noteq 3 then jmp START2\n\
> # The elements 1 and 2 have the orders 2 and 3, respectively.\n\
> set X 0\n\
> lbl CONJ\n\
> incr X\n\
> if X gt 100 then timeout\n\
> rand 3\n\
> cjr 2 3\n\
> mu 1 2 4 # ab\n\
> ord 4 C\n\
> if C notin 2 3 5 then fail\n\
> if C noteq 5 then jmp CONJ\n\
> oup 2 1 2";;
gap> find:= ScanBBoxProgram( findstr );
rec( program := <black box program> )
</pre></td></tr></table>
<p>The second example is a black box program that checks whether its two inputs are standard generators for A_5.</p>
<table class="example">
<tr><td><pre>
gap> checkstr:= "\
> chor 1 2\n\
> chor 2 3\n\
> mu 1 2 3\n\
> chor 3 5";;
gap> check:= ScanBBoxProgram( checkstr );
rec( program := <black box program> )
</pre></td></tr></table>
<p><a id="X7D211A5D8602B330" name="X7D211A5D8602B330"></a></p>
<h5>4.2-3 RunBBoxProgram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> RunBBoxProgram</code>( <var class="Arg">prog, G, input, options</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a record describing the result and the statistics of running the black box program <var class="Arg">prog</var>, or <code class="keyw">fail</code>, or the string <code class="code">"timeout"</code>.</p>
<p>For a black box program <var class="Arg">prog</var>, a group <var class="Arg">G</var>, a list <var class="Arg">input</var> of group elements, and a record <var class="Arg">options</var>, <code class="func">RunBBoxProgram</code> applies <var class="Arg">prog</var> to <var class="Arg">input</var>, where <var class="Arg">G</var> is used only to compute random elements.</p>
<p>The return value is <code class="keyw">fail</code> if a syntax error or an explicit <code class="code">fail</code> statement is reached at runtime, and the string <code class="code">"timeout"</code> if a <code class="code">timeout</code> statement is reached. (The latter might mean that the random choices were unlucky.) Otherwise a record with the following components is returned.</p>
<dl>
<dt><strong class="Mark"><code class="code">gens</code></strong></dt>
<dd><p>a list of group elements, bound if an <code class="code">oup</code> statement was reached,</p>
</dd>
<dt><strong class="Mark"><code class="code">result</code></strong></dt>
<dd><p><code class="keyw">true</code> if a <code class="code">true</code> statement was reached, <code class="keyw">false</code> if either a <code class="code">false</code> statement or a failed order check was reached,</p>
</dd>
</dl>
<p>The other components serve as statistical information about the numbers of the various operations (<code class="code">multiply</code>, <code class="code">invert</code>, <code class="code">power</code>, <code class="code">order</code>, <code class="code">random</code>, <code class="code">conjugate</code>, <code class="code">conjugateinplace</code>, <code class="code">commutator</code>), and the runtime in milliseconds (<code class="code">timetaken</code>).</p>
<p>The following components of <var class="Arg">options</var> are supported.</p>
<dl>
<dt><strong class="Mark"><code class="code">randomfunction</code></strong></dt>
<dd><p>the function called with argument <var class="Arg">G</var> in order to compute a random element of <var class="Arg">G</var> (default <code class="func">PseudoRandom</code> (<a href="../../../doc/htm/ref/CHAP028.htm#SECT006"><b>Reference: PseudoRandom</b></a>))</p>
</dd>
<dt><strong class="Mark"><code class="code">orderfunction</code></strong></dt>
<dd><p>the function for computing element orders (the default is <code class="func">Order</code> (<a href="../../../doc/htm/ref/CHAP030.htm#SECT010"><b>Reference: Order</b></a>)),</p>
</dd>
<dt><strong class="Mark"><code class="code">quiet</code></strong></dt>
<dd><p>ignore <code class="code">echo</code> statements (default <code class="keyw">false</code>),</p>
</dd>
<dt><strong class="Mark"><code class="code">verbose</code></strong></dt>
<dd><p>print information about the line that is currently processed, and about order checks (default <code class="keyw">false</code>),</p>
</dd>
<dt><strong class="Mark"><code class="code">allowbreaks</code></strong></dt>
<dd><p>call <code class="func">Error</code> (<a href="../../../doc/htm/ref/CHAP006.htm#SECT006"><b>Reference: Error</b></a>) when a <code class="code">break</code> statement is reached (default <code class="keyw">true</code>).</p>
</dd>
</dl>
<p>As an example, we run the black box programs constructed in the example for <code class="func">ScanBBoxProgram</code> (<a href="chap4.html#X7EA20532868F9863"><b>4.2-2</b></a>).</p>
<table class="example">
<tr><td><pre>
gap> g:= AlternatingGroup( 5 );;
gap> res:= RunBBoxProgram( find.program, g, [], rec() );;
gap> IsBound( res.gens ); IsBound( res.result );
true
false
gap> List( res.gens, Order );
[ 2, 3 ]
gap> Order( Product( res.gens ) );
5
gap> res:= RunBBoxProgram( check.program, "dummy", res.gens, rec() );;
gap> IsBound( res.gens ); IsBound( res.result );
false
true
gap> res.result;
true
gap> othergens:= GeneratorsOfGroup( g );;
gap> res:= RunBBoxProgram( check.program, "dummy", othergens, rec() );;
gap> res.result;
false
</pre></td></tr></table>
<p><a id="X869BACFB80A3CC87" name="X869BACFB80A3CC87"></a></p>
<h5>4.2-4 ResultOfBBoxProgram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> ResultOfBBoxProgram</code>( <var class="Arg">prog, G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a list of group elements or <code class="keyw">true</code>, <code class="keyw">false</code>, <code class="keyw">fail</code>, or the string <code class="code">"timeout"</code>.</p>
<p>This function calls <code class="func">RunBBoxProgram</code> (<a href="chap4.html#X7D211A5D8602B330"><b>4.2-3</b></a>) with the black box program <var class="Arg">prog</var> and second argument either a group or a list of group elements; the default options are assumed. The return value is <code class="keyw">fail</code> if this call yields <code class="keyw">fail</code>, otherwise the <code class="code">gens</code> component of the result, if bound, or the <code class="code">result</code> component if not.</p>
<p>As an example, we run the black box programs constructed in the example for <code class="func">ScanBBoxProgram</code> (<a href="chap4.html#X7EA20532868F9863"><b>4.2-2</b></a>).</p>
<table class="example">
<tr><td><pre>
gap> g:= AlternatingGroup( 5 );;
gap> res:= ResultOfBBoxProgram( find.program, g );;
gap> List( res, Order );
[ 2, 3 ]
gap> Order( Product( res ) );
5
gap> res:= ResultOfBBoxProgram( check.program, res );
true
gap> othergens:= GeneratorsOfGroup( g );;
gap> res:= ResultOfBBoxProgram( check.program, othergens );
false
</pre></td></tr></table>
<p><a id="X826ACFE887E0B6B8" name="X826ACFE887E0B6B8"></a></p>
<h5>4.2-5 AsBBoxProgram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AsBBoxProgram</code>( <var class="Arg">slp</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>an equivalent black box program for the given straight line program or straight line decision.</p>
<p>Let <var class="Arg">slp</var> be a straight line program (see <code class="func">IsStraightLineProgram</code> (<a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: IsStraightLineProgram</b></a>)) or a straight line decision (see <code class="func">IsStraightLineDecision</code> (<a href="chap4.html#X8787E2EC7DB85A89"><b>4.1-1</b></a>)). Then <code class="func">AsBBoxProgram</code> returns a black box program <var class="Arg">bbox</var> (see <code class="func">IsBBoxProgram</code> (<a href="chap4.html#X87CAF2DE870D0E3B"><b>4.2-1</b></a>)) with the "same" output as <var class="Arg">slp</var>, in the sense that <code class="func">ResultOfBBoxProgram</code> (<a href="chap4.html#X869BACFB80A3CC87"><b>4.2-4</b></a>) yields the same result for <var class="Arg">bbox</var> as <code class="func">ResultOfStraightLineProgram</code> (<a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: ResultOfStraightLineProgram</b></a>) or <code class="func">ResultOfStraightLineDecision</code> (<a href="chap4.html#X7E7B328A84685480"><b>4.1-6</b></a>), respectively, for <var class="Arg">slp</var>.</p>
<table class="example">
<tr><td><pre>
gap> f:= FreeGroup( "x", "y" );; gens:= GeneratorsOfGroup( f );;
gap> slp:= StraightLineProgram( [ [1,2,2,3], [3,-1] ], 2 );
<straight line program>
gap> ResultOfStraightLineProgram( slp, gens );
y^-3*x^-2
gap> bboxslp:= AsBBoxProgram( slp );
<black box program>
gap> ResultOfBBoxProgram( bboxslp, gens );
[ y^-3*x^-2 ]
gap> lines:= [ [ "Order", 1, 2 ], [ "Order", 2, 3 ],
> [ [ 1, 1, 2, 1 ], 3 ], [ "Order", 3, 5 ] ];;
gap> dec:= StraightLineDecision( lines, 2 );
<straight line decision>
gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,3,5) ] );
true
gap> ResultOfStraightLineDecision( dec, [ (1,2)(3,4), (1,3,4) ] );
false
gap> bboxdec:= AsBBoxProgram( dec );
<black box program>
gap> ResultOfBBoxProgram( bboxdec, [ (1,2)(3,4), (1,3,5) ] );
true
gap> ResultOfBBoxProgram( bboxdec, [ (1,2)(3,4), (1,3,4) ] );
false
</pre></td></tr></table>
<p><a id="X7D36DFA87C8B2C48" name="X7D36DFA87C8B2C48"></a></p>
<h5>4.2-6 AsStraightLineProgram</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AsStraightLineProgram</code>( <var class="Arg">bbox</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><b>Returns: </b>an equivalent straight line program for the given black box program, or <code class="keyw">fail</code>.</p>
<p>For a black box program (see <code class="func">AsBBoxProgram</code> (<a href="chap4.html#X826ACFE887E0B6B8"><b>4.2-5</b></a>)) <var class="Arg">bbox</var>, <code class="func">AsStraightLineProgram</code> returns a straight line program (see <code class="func">IsStraightLineProgram</code> (<a href="../../../doc/htm/ref/CHAP035.htm#SECT008"><b>Reference: IsStraightLineProgram</b></a>)) with the same output as <var class="Arg">bbox</var> if such a straight line program exists, and <code class="keyw">fail</code> otherwise.</p>
<table class="example">
<tr><td><pre>
gap> Display( AsStraightLineProgram( bboxslp ) );
# input:
r:= [ g1, g2 ];
# program:
r[3]:= r[1]^2;
r[4]:= r[2]^3;
r[5]:= r[3]*r[4];
r[3]:= r[5]^-1;
# return values:
[ r[3] ]
gap> AsStraightLineProgram( bboxdec );
fail
</pre></td></tr></table>
<p><a id="X87E1F08D80C9E069" name="X87E1F08D80C9E069"></a></p>
<h4>4.3 <span class="Heading">Representations of Minimal Degree</span></h4>
<p>This section deals with minimal degrees of permutation and matrix representations. We do not provide an algorithm that computes these degrees for an arbitrary group, we only provide some tools for evaluating known databases, mainly concerning nearly simple groups, in order to derive the minimal degrees, see Section <a href="chap4.html#X7FC33DFF8481F8D1"><b>4.3-4</b></a>.</p>
<p>In the <strong class="pkg">AtlasRep</strong> package, this information is used in <code class="func">DisplayAtlasInfo</code> (<a href="chap2.html#X79DACFFA7E2D1A99"><b>2.5-1</b></a>), <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap2.html#X841478AB7CD06D44"><b>2.5-4</b></a>), and <code class="func">AllAtlasGeneratingSetInfos</code> (<a href="chap2.html#X84C2D76482E60E42"><b>2.5-5</b></a>).</p>
<p><a id="X7DC66D8282B2BB7F" name="X7DC66D8282B2BB7F"></a></p>
<h5>4.3-1 MinimalRepresentationInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> MinimalRepresentationInfo</code>( <var class="Arg">grpname, conditions</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a record with the components <code class="code">value</code> and <code class="code">source</code>, or <code class="keyw">fail</code></p>
<p>Let <var class="Arg">groupname</var> be the <strong class="pkg">GAP</strong> name of a group G, say. If the information described by <var class="Arg">conditions</var> about minimal representations of this group can be computed or is stored then <code class="func">MinimalRepresentationInfo</code> returns a record with the components <code class="code">value</code> and <code class="code">source</code>, otherwise <code class="keyw">fail</code> is returned.</p>
<p>The following values for <var class="Arg">conditions</var> are supported.</p>
<ul>
<li><p>If <var class="Arg">conditions</var> is <code class="func">NrMovedPoints</code> (<a href="../../../doc/htm/ref/CHAP040.htm#SECT002"><b>Reference: NrMovedPoints</b></a>) then <code class="code">value</code>, if known, is the degree of a minimal faithful permutation representation for G.</p>
</li>
<li><p>If <var class="Arg">conditions</var> consists of <code class="func">Characteristic</code> (<a href="../../../doc/htm/ref/CHAP030.htm#SECT010"><b>Reference: Characteristic</b></a>) and a prime integer <var class="Arg">p</var> then <code class="code">value</code>, if known, is the dimension of a minimal faithful matrix representation in characteristic <var class="Arg">p</var> for <var class="Arg">G</var>.</p>
</li>
<li><p>If <var class="Arg">conditions</var> consists of <code class="func">Size</code> (<a href="../../../doc/htm/ref/CHAP028.htm#SECT003"><b>Reference: Size</b></a>) and a prime power <var class="Arg">q</var> then <code class="code">value</code>, if known, is the dimension of a minimal faithful matrix representation over the field of size <var class="Arg">q</var> for <var class="Arg">G</var>.</p>
</li>
</ul>
<p>In all cases, the value of the component <code class="code">source</code> is a list of strings that describe sources of the information, which can be the ordinary or modular character table of <code class="code">G</code> (see <a href="chapBib.html#biBCCN85">[CCNPW85]</a>, <a href="chapBib.html#biBJLPW95">[JLPW95]</a>, <a href="chapBib.html#biBHL89">[HL89]</a>), the table of marks of <code class="code">G</code>, or <a href="chapBib.html#biBJan05">[Jan05]</a>. For an overview of minimal degrees of faithful matrix representations for sporadic simple groups and their covering groups, see also</p>
<p><span class="URL"><a href="http://www.math.rwth-aachen.de/~MOC/mindeg/">http://www.math.rwth-aachen.de/~MOC/mindeg/</a></span>.</p>
<p>Note that this function does not give any information about minimal representations over prescribed fields in characteristic zero.</p>
<p>Information about groups that occur in the <strong class="pkg">AtlasRep</strong> package is precomputed in <code class="func">MinimalRepresentationInfoData</code> (<a href="chap4.html#X7E1B76DC86A8C405"><b>4.3-2</b></a>), so the packages <strong class="pkg">CTblLib</strong> and <strong class="pkg">TomLib</strong> are not needed when <code class="func">MinimalRepresentationInfo</code> is called for these groups. (The only case that is not covered by this list is that one asks for the minimal degree of matrix representations over a prescribed field in characteristic coprime to the group order.)</p>
<p>One of the following strings can be given as an additional last argument.</p>
<dl>
<dt><strong class="Mark"><code class="code">"cache"</code></strong></dt>
<dd><p>means that the function tries to compute (and then store) values that are not stored in <code class="func">MinimalRepresentationInfoData</code> (<a href="chap4.html#X7E1B76DC86A8C405"><b>4.3-2</b></a>), but stored values are preferred; this is also the default.</p>
</dd>
<dt><strong class="Mark"><code class="code">"lookup"</code></strong></dt>
<dd><p>means that stored values are returned but the function does not attempt to compute values that are not stored in <code class="func">MinimalRepresentationInfoData</code> (<a href="chap4.html#X7E1B76DC86A8C405"><b>4.3-2</b></a>).</p>
</dd>
<dt><strong class="Mark"><code class="code">"recompute"</code></strong></dt>
<dd><p>means that the function always tries to compute the desired value, and checks the result against stored values.</p>
</dd>
</dl>
<table class="example">
<tr><td><pre>
gap> MinimalRepresentationInfo( "A5", NrMovedPoints );
rec( value := 5,
source := [ "computed (alternating group)", "computed (char. table)",
"computed (subgroup tables)",
"computed (subgroup tables, known repres.)",
"computed (table of marks)" ] )
gap> MinimalRepresentationInfo( "A5", Characteristic, 2 );
rec( value := 2, source := [ "computed (char. table)" ] )
gap> MinimalRepresentationInfo( "A5", Size, 2 );
rec( value := 4, source := [ "computed (char. table)" ] )
</pre></td></tr></table>
<p><a id="X7E1B76DC86A8C405" name="X7E1B76DC86A8C405"></a></p>
<h5>4.3-2 MinimalRepresentationInfoData</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> MinimalRepresentationInfoData</code></td><td class="tdright">( global variable )</td></tr></table></div>
<p>This is a record whose components are <strong class="pkg">GAP</strong> names of groups for which information about minimal permutation and matrix representations were known in advance or have been computed in the current <strong class="pkg">GAP</strong> session. The value for the group G, say, is a record with the following components.</p>
<dl>
<dt><strong class="Mark"><code class="code">NrMovedPoints</code></strong></dt>
<dd><p>a record with the components <code class="code">value</code> (the degree of a smallest faithful permutation representation of G) and <code class="code">source</code> (a string describing the source of this information).</p>
</dd>
<dt><strong class="Mark"><code class="code">Characteristic</code></strong></dt>
<dd><p>a record whose components are at most <code class="code">0</code> and strings corresponding to prime integers, each bound to a record with the components <code class="code">value</code> (the degree of a smallest faithful matrix representation of G in this characteristic) and <code class="code">source</code> (a string describing the source of this information).</p>
</dd>
<dt><strong class="Mark"><code class="code">CharacteristicAndSize</code></strong></dt>
<dd><p>a record whose components are strings corresponding to prime integers <var class="Arg">p</var>, each bound to a record with the components <code class="code">sizes</code> (a list of powers <var class="Arg">q</var> of <var class="Arg">p</var>), <code class="code">dimensions</code> (the corresponding list of minimal dimensions of faithful matrix representations of G over a field of size <var class="Arg">q</var>), <code class="code">sources</code> (the corresponding list of strings describing the source of this information), and <code class="code">complete</code> (a record with the components <code class="code">val</code> (<code class="keyw">true</code> if the minimal dimension over <em>any</em> finite field in characteristic <var class="Arg">p</var> can be derived from the values in the record, and <code class="keyw">false</code> otherwise) and <code class="code">source</code> (a string describing the source of this information)).</p>
</dd>
</dl>
<p>The values are set by <code class="func">SetMinimalRepresentationInfo</code> (<a href="chap4.html#X79C4C9F683E919C9"><b>4.3-3</b></a>).</p>
<p><a id="X79C4C9F683E919C9" name="X79C4C9F683E919C9"></a></p>
<h5>4.3-3 SetMinimalRepresentationInfo</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> SetMinimalRepresentationInfo</code>( <var class="Arg">grpname, op, value, source</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b><code class="keyw">true</code> if the values were successfully set, <code class="keyw">false</code> if stored values contradict the given ones.</p>
<p>This function sets an entry in <code class="func">MinimalRepresentationInfoData</code> (<a href="chap4.html#X7E1B76DC86A8C405"><b>4.3-2</b></a>) for the group G, say, with <strong class="pkg">GAP</strong> name <var class="Arg">grpname</var>.</p>
<p>Supported values for <var class="Arg">op</var> are</p>
<ul>
<li><p><code class="code">"NrMovedPoints"</code> (see <code class="func">NrMovedPoints</code> (<a href="../../../doc/htm/ref/CHAP040.htm#SECT002"><b>Reference: NrMovedPoints</b></a>)), which means that <var class="Arg">value</var> is the degree of minimal faithful permutation representations of G,</p>
</li>
<li><p>a list of length two with first entry <code class="code">"Characteristic"</code> (see <code class="func">Characteristic</code> (<a href="../../../doc/htm/ref/CHAP030.htm#SECT010"><b>Reference: Characteristic</b></a>)) and second entry <var class="Arg">char</var> either zero or a prime integer, which means that <var class="Arg">value</var> is the dimension of minimal faithful matrix representations of G in characteristic <var class="Arg">char</var>,</p>
</li>
<li><p>a list of length two with first entry <code class="code">"Size"</code> (see <code class="func">Size</code> (<a href="../../../doc/htm/ref/CHAP028.htm#SECT003"><b>Reference: Size</b></a>)) and second entry a prime power <var class="Arg">q</var>, which means that <var class="Arg">value</var> is the dimension of minimal faithful matrix representations of G over the field with <var class="Arg">q</var> elements, and</p>
</li>
<li><p>a list of length three with first entry <code class="code">"Characteristic"</code> (see <code class="func">Characteristic</code> (<a href="../../../doc/htm/ref/CHAP030.htm#SECT010"><b>Reference: Characteristic</b></a>)), second entry a prime integer <var class="Arg">p</var>, and third entry the string <code class="code">"complete"</code>, which means that the information stored for characteristic <var class="Arg">p</var> is complete in the sense that for any given power q of <var class="Arg">p</var>, the minimal faithful degree over the field with q elements equals that for the largest stored field size of which q is a power.</p>
</li>
</ul>
<p>In each case, <var class="Arg">source</var> is a string describing the source of the data; <em>computed</em> values are detected from the prefix <code class="code">"comp"</code> of <var class="Arg">source</var>.</p>
<p>If the intended value is already stored and differs from <var class="Arg">value</var> then an error message is printed.</p>
<table class="example">
<tr><td><pre>
gap> SetMinimalRepresentationInfo( "A5", "NrMovedPoints", 5,
> "computed (alternating group)" );
true
gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 0 ], 3,
> "computed (char. table)" );
true
gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 2 ], 2,
> "computed (char. table)" );
true
gap> SetMinimalRepresentationInfo( "A5", [ "Size", 2 ], 4,
> "computed (char. table)" );
true
gap> SetMinimalRepresentationInfo( "A5", [ "Size", 4 ], 2,
> "computed (char. table)" );
true
gap> SetMinimalRepresentationInfo( "A5", [ "Characteristic", 3 ], 3,
> "computed (char. table)" );
true
</pre></td></tr></table>
<p><a id="X7FC33DFF8481F8D1" name="X7FC33DFF8481F8D1"></a></p>
<h5>4.3-4 <span class="Heading">Criteria Used to Compute Minimality Information</span></h5>
<p>Let <var class="Arg">grpname</var> be the <strong class="pkg">GAP</strong> name of a group G, say.</p>
<p>The information about the minimal degree of a faithful <em>matrix representation</em> of G in a given characteristic or over a given field in positive characteristic is derived from the relevant (ordinary or modular) character table of G, except in a few cases where this table itself is not known but enough information about the degrees is available in <a href="chapBib.html#biBHL89">[HL89]</a> and <a href="chapBib.html#biBJan05">[Jan05]</a>.</p>
<p>The following criteria are used for deriving the minimal degree of a faithful <em>permutation representation</em> of G from the information in the <strong class="pkg">GAP</strong> libraries of character tables and of tables of marks.</p>
<ul>
<li><p>If <var class="Arg">grpname</var> has the form <code class="code">A<var class="Arg">n</var></code> or <code class="code">A<var class="Arg">n</var>.2</code> (denoting alternating and symmetric groups, respectively) then the minimal degree is <var class="Arg">n</var>, except if <var class="Arg">n</var> is smaller than 3 or 2, respectively.</p>
</li>
<li><p>If <var class="Arg">grpname</var> has the form <code class="code">L2(<var class="Arg">q</var>)</code> (denoting projective special linear groups in dimension two) then the minimal degree is <var class="Arg">q</var> + 1, except if <var class="Arg">q</var> in { 2, 3, 5, 7, 9, 11 }, see <a href="chapBib.html#biBHup67">[Hup67, Satz II.8.28]</a>.</p>
</li>
<li><p>If the largest maximal subgroup of G is core-free then the index of this subgroup is the minimal degree. (This is used when the two character tables in question and the class fusion are available in the <strong class="pkg">GAP</strong> Character Table Library; this happens for many character tables of simple groups.)</p>
</li>
<li><p>If G has a unique minimal normal subgroup then each minimal faithful permutation representation is transitive.</p>
<p>In this case, the minimal degree can be computed directly from the information in the table of marks of G if this is available in <strong class="pkg">GAP</strong>'s library of tables of marks.</p>
<p>Suppose that the largest maximal subgroup of G is not core-free but simple and normal in G, and that the other maximal subgroups of G are core-free. In this case, we take the minimum of the indices of the core-free maximal subgroups and of the product of index and minimal degree of the normal maximal subgroup. (This suffices since no core-free subgroup of the whole group can contain a nontrivial normal subgroup of a normal maximal subgroup.)</p>
<p>Let N be the unique minimal normal subgroup of G, and assume that G/N is simple and has minimal degree n, say. If there is a subgroup U of index n * |N| in G that intersects N trivially then the minimal degree of G is n * |N|. (This is used for the case that N is central in G and N x U occurs as a subgroup of G.)</p>
</li>
<li><p>If we know a subgroup of G whose minimal degree is <var class="Arg">n</var>, say, and if we know either (a class fusion from) a core-free subgroup of index <var class="Arg">n</var> in G or a faithful permutation representation of degree <var class="Arg">n</var> for G then <var class="Arg">n</var> is the minimal degree for G. (This happens often for tables of almost simple groups.)</p>
</li>
</ul>
<p><a id="X85217A9578DA034A" name="X85217A9578DA034A"></a></p>
<h5>4.3-5 AGR_TestMinimalDegrees</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> AGR_TestMinimalDegrees</code>( <var class="Arg"></var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b><code class="keyw">true</code> if no contradiction was found, and <code class="keyw">false</code> otherwise.</p>
<p>This function checks that the (permutation and matrix) representations available in the <strong class="pkg">ATLAS</strong> of group representations do not have smaller degree than the claimed minimum.</p>
<p>An error message is printed for each contradiction found.</p>
<p><a id="X7F31A7CB841FE63F" name="X7F31A7CB841FE63F"></a></p>
<h5>4.3-6 BrowseMinimalDegrees</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> BrowseMinimalDegrees</code>( <var class="Arg">[groupnames]</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>the list of info records for the clicked representations.</p>
<p>If the <strong class="pkg">GAP</strong> package <strong class="pkg">Browse</strong> (see <a href="chapBib.html#biBBrowse1.2">[BL08]</a>) is loaded then the function <code class="func">BrowseMinimalDegrees</code> is available. It opens a browse table whose rows correspond to the groups for which the <strong class="pkg">ATLAS</strong> of Group Representations contains some information about minimal degrees, whose columns correspond to the characteristics that occur, and whose entries are the known minimal degrees.</p>
<table class="example">
<tr><td><pre>
gap> if LoadPackage( "browse", "1.2" ) = true then
> down:= NCurses.keys.DOWN;; DOWN:= NCurses.keys.NPAGE;;
> right:= NCurses.keys.RIGHT;; END:= NCurses.keys.END;;
> enter:= NCurses.keys.ENTER;; nop:= [ 14, 14, 14 ];;
> # just scroll in the table
> BrowseData.SetReplay( Concatenation( [ DOWN, DOWN, DOWN,
> right, right, right ], "sedddrrrddd", nop, nop, "Q" ) );
> BrowseMinimalDegrees();;
> # restrict the table to the groups with minimal ordinary degree 6
> BrowseData.SetReplay( Concatenation( "scf6",
> [ down, down, right, enter, enter ] , nop, nop, "Q" ) );
> BrowseMinimalDegrees();;
> BrowseData.SetReplay( false );
> fi;
</pre></td></tr></table>
<p>If an argument <var class="Arg">groupnames</var> is given then it must be a list of group names of the <strong class="pkg">ATLAS</strong> of Group Representations; the browse table is then restricted to the rows corresponding to these group names and to the columns that are relevant for these groups. A perhaps interesting example is the subtable with the data concerning sporadic simple groups and their covering groups, which has been published in <a href="chapBib.html#biBJan05">[Jan05]</a>. This table can be shown as follows.</p>
<table class="example">
<tr><td><pre>
gap> if LoadPackage( "browse", "1.2" ) = true then
> # just scroll in the table
> BrowseData.SetReplay( Concatenation( [ DOWN, DOWN, DOWN, END ],
> "rrrrrrrrrrrrrr", nop, nop, "Q" ) );
> BrowseMinimalDegrees( BibliographySporadicSimple.groupNamesJan05 );;
> fi;
</pre></td></tr></table>
<p>(The browse table does not contain rows for the groups 6.M_22, 12.M_22, 6.Fi_22. Note that in spite of the title of <a href="chapBib.html#biBJan05">[Jan05]</a>, the entries in Table 1 of this paper are in fact the minimal degrees of faithful <em>irreducible</em> representations, and in the above three cases, these degrees are larger than the minimal degrees of faithful representations. The underlying data of the browse table is about the minimal faithful degrees.)</p>
<p>The return value of <code class="func">BrowseMinimalDegrees</code> is the list of <code class="func">OneAtlasGeneratingSetInfo</code> (<a href="chap2.html#X841478AB7CD06D44"><b>2.5-4</b></a>) values for those representations that have been "clicked" in visual mode.</p>
<p>The variant without arguments of this function is also available in the menu shown by <code class="func">BrowseGapData</code> (<a href="../../../pkg/Browse/doc/chap6.html#X850C786C87A4877B"><b>Browse: BrowseGapData</b></a>).</p>
<p><a id="X7DD6CE0E79AAD61B" name="X7DD6CE0E79AAD61B"></a></p>
<h4>4.4 <span class="Heading">Bibliographies of Sporadic Simple Groups</span></h4>
<p>The bibliographies contained in the <strong class="pkg">ATLAS</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCNPW85]</a> and in the <strong class="pkg">ATLAS</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a> are available online in HTML format, see <span class="URL"><a href="http://www.gap-system.org/Manuals/pkg/atlasrep/bibl/index.html">http://www.gap-system.org/Manuals/pkg/atlasrep/bibl/index.html</a></span>.</p>
<p>The source data in BibXMLext format is part of the <strong class="pkg">AtlasRep</strong> package, in four files with suffix <code class="file">xml</code> in the package's <code class="file">bibl</code> directory. Note that each of the two books contains two bibliographies.</p>
<p>Details about the BibXMLext format, including information how to transform the data into other formats such as BibTeX, can be found in the <strong class="pkg">GAP</strong> package <strong class="pkg">GAPDoc</strong> (see <a href="chapBib.html#biBGAPDoc">[LN08]</a>).</p>
<p>These source files are used also by the function <code class="func">BrowseBibliographySporadicSimple</code> (<a href="chap4.html#X84ED4FC182C28198"><b>4.4-1</b></a>).</p>
<p><a id="X84ED4FC182C28198" name="X84ED4FC182C28198"></a></p>
<h5>4.4-1 BrowseBibliographySporadicSimple</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> BrowseBibliographySporadicSimple</code>( <var class="Arg"></var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><b>Returns: </b>a record as returned by <code class="func">ParseBibXMLExtString</code> (<a href="../../../pkg/gapdoc/doc/chap7.html#X86BD29AE7A453721"><b>GAPDoc: ParseBibXMLextString</b></a>).</p>
<p>If the <strong class="pkg">GAP</strong> package <strong class="pkg">Browse</strong> (see <a href="chapBib.html#biBBrowse1.2">[BL08]</a>) is loaded then this function is available. It opens a browse table whose rows correspond to the entries of the bibliographies of the <strong class="pkg">ATLAS</strong> of Finite Groups <a href="chapBib.html#biBCCN85">[CCNPW85]</a> and the <strong class="pkg">ATLAS</strong> of Brauer Characters <a href="chapBib.html#biBJLPW95">[JLPW95]</a>.</p>
<p>The function is based on <code class="func">BrowseBibliography</code> (<a href="../../../pkg/Browse/doc/chap6.html#X7F0FE4CC7F46ABF3"><b>Browse: BrowseBibliography</b></a>), see the documentation of this function for details, e.g., about the return value.</p>
<p>The returned record encodes the bibliography entries corresponding to those rows of the table that are "clicked" in visual mode, in the same format as the return value of <code class="func">ParseBibXMLExtString</code> (<a href="../../../pkg/gapdoc/doc/chap7.html#X86BD29AE7A453721"><b>GAPDoc: ParseBibXMLextString</b></a>), see the manual of the <strong class="pkg">GAP</strong> package <strong class="pkg">GAPDoc</strong> <a href="chapBib.html#biBGAPDoc">[LN08]</a> for details.</p>
<p><code class="func">BrowseBibliographySporadicSimple</code> can be called also via the menu shown by <code class="func">BrowseGapData</code> (<a href="../../../pkg/Browse/doc/chap6.html#X850C786C87A4877B"><b>Browse: BrowseGapData</b></a>).</p>
<table class="example">
<tr><td><pre>
gap> if LoadPackage( "browse", "1.2" ) = true then
> enter:= NCurses.keys.ENTER;; nop:= [ 14, 14, 14 ];;
> BrowseData.SetReplay( Concatenation(
> # choose the application
> "/Bibliography of Sporadic Simple Groups", [ enter, enter ],
> # search in the title column for the Atlas of Finite Groups
> "scr/Atlas of finite groups", [ enter,
> # and quit
> nop, nop, nop, nop ], "Q" ) );
> BrowseGapData();;
> BrowseData.SetReplay( false );
> fi;
</pre></td></tr></table>
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