<html><head><title>[GrpConst] 6 The Upwards Extension Method</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP005.htm">Previous</a>] [<a href ="CHAP007.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <h1>6 The Upwards Extension Method</h1><p> <p> <a name = "I0"></a> This is a method to construct up to isomorphism the finite groups of a given order. For this purpose it will loop over all possible perfect groups and construct upwards extensions by soluble groups. This, in turn, is done by iterated cyclic extensions. <p> Since this method is less efficient than the above two methods, it will usually only be used for the determination of non-soluble groups. <p> <a name = ""></a> <li><code>UpwardsExtensions( </code><var>G</var><code>, </code><var>s</var><code> ) F</code> <p> Let <var>G</var> be a permutation group and <var>s</var> a positive integer. This function returns a list corresponding to <code>DivisorsInt(s)</code>. Let <var>t</var> be the <var>i</var>-th divisor of <var>s</var>. Then the <var>i</var>-th entry in the output is a list of all extensions of <var>G</var> by a soluble group of order <var>t</var> up to isomorphism. The returned groups are permutation groups again. <p> Typically, this function is applied to perfect groups <var>G</var>, which may be obtained from the perfect groups catalogue in GAP (see the Section on <code>Finite perfect groups</code> in the reference manual). <p> The most time-consuming part of the computation in <code>UpwardsExtensions</code> is the isomorphism test. The following function does no reduction to isomorphism type representatives and hence is much more efficient. <p> <a name = ""></a> <li><code>CyclicExtensions( </code><var>G</var><code>, </code><var>p</var><code> ) F</code> <p> Here <var>G</var> should be a permutation group and <var>p</var> a prime. This function computes a list of permutation groups containing the upwards extensions of <var>G</var> by the cyclic group of order <var>p</var>, but not reduced to isomorphism type representatives. <p> There is an info class <code>InfoUpExt</code> available with values from 1 to 3. <p> <pre> gap> G := PerfectGroup( IsPermGroup, 120, 1 ); A5 2^1 gap> c := CyclicExtensions( G, 2 );; gap> List( c, IdGroup ); [ [ 240, 94 ], [ 240, 93 ], [ 240, 90 ], [ 240, 89 ] ] gap> H := c[1]; <permutation group of size 240 with 2 generators> gap> CyclicExtensions( H, 2 );; gap> List(last, IdGroup); [ [ 480, 960 ], [ 480, 955 ], [ 480, 222 ], [ 480, 222 ], [ 480, 953 ], [ 480, 953 ], [ 480, 957 ], [ 480, 957 ], [ 480, 949 ], [ 480, 950 ], [ 480, 219 ], [ 480, 219 ] ] gap> u := UpwardsExtensions( G, 4 );; gap> List( u, Length ); [ 1, 4, 14 ] gap> List( u[3], IdGroup); [ [ 480, 960 ], [ 480, 959 ], [ 480, 950 ], [ 480, 222 ], [ 480, 221 ], [ 480, 947 ], [ 480, 949 ], [ 480, 219 ], [ 480, 948 ], [ 480, 218 ], [ 480, 955 ], [ 480, 957 ], [ 480, 953 ], [ 480, 946 ] ] </pre> <p> In case that we want to extend a perfect group with trivial centre, then there is a better algorithm available. This is implemented as well and can be used with the following functions. <p> <a name = ""></a> <li><code>UpwardsExtensionsNoCentre( </code><var>G</var><code>, </code><var>s</var><code> ) F</code> <p> Let <var>G</var> be a perfect permutation group with trivial centre and <var>s</var> a positive integer. This function returns a list of all extensions of <var>G</var> by a soluble group of order <var>s</var> up to isomorphism. The returned groups are permutation groups again. Note that, in difference to <code>UpwardsExtensions</code> this function does not return the extensions by groups of order dividing <var>s</var>. Moreover, the implementation of the function requires that all soluble groups of order <var>s</var> are available as <code>SmallGroups</code>. The implementation then uses the following function to determine groups. <p> <a name = ""></a> <li><code>ExtensionsByGroupNoCentre( </code><var>G</var><code>, </code><var>H</var><code> ) F</code> <p> Let <var>G</var> be a perfect permutation group with trivial centre and <var>H</var> a soluble group. This functions returns all extensions of <var>G</var> by <var>H</var> up to isomorphism. <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP005.htm">Previous</a>] [<a href ="CHAP007.htm">Next</a>] [<a href = "theindex.htm">Index</a>] <P> <address>GrpConst manual<br>August 2003 </address></body></html>