<html><head><title>[CRISP] 6 Examples of group classes</title></head> <body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP005.htm">Previous</a>] [<a href = "theindex.htm">Index</a>] <h1>6 Examples of group classes</h1><p> <P> <H3>Sections</H3> <oL> <li> <A HREF="CHAP006.htm#SECT001">Pre-defined group classes</a> <li> <A HREF="CHAP006.htm#SECT002">Pre-defined projector functions</a> <li> <A HREF="CHAP006.htm#SECT003">Pre-defined sets of primes</a> </ol><p> <p> This chapter describes some pre-defined group classes, namely the classes of all abelian, nilpotent, and supersolvable groups. Moreover, there are some functions constructing the classes of all <var>p</var>-groups, <var>pi</var>-groups, and abelian groups whose exponent divides a given positive integer. <p> The definitions of these group classes can also serve as further examples of how group classes can be defined using the methods described in the preceding chapters. <p> <p> <h2><a name="SECT001">6.1 Pre-defined group classes</a></h2> <p><p> <a name = "SSEC001.1"></a> <li><code>TrivialGroups V</code> <p> <a name = "I0"></a> <a name = "I0"></a> <a name = "I1"></a> The global variable <code>TrivialGroups</code> contains the class of all trivial groups. It is a subgroup closed saturated Fitting formation. <p> <a name = "SSEC001.2"></a> <li><code>NilpotentGroups V</code> <p> <a name = "I2"></a> <a name = "I2"></a> <a name = "I3"></a> subgroup closed saturated Fitting formation. <p> <a name = "SSEC001.3"></a> <li><code>SupersolvableGroups V</code> <p> <a name = "I4"></a> <a name = "I4"></a> <a name = "I5"></a> is a subgroup closed saturated formation. <p> <a name = "SSEC001.4"></a> <li><code>AbelianGroups V</code> <p> <a name = "I6"></a> <a name = "I7"></a> <a name = "I7"></a> <a name = "I8"></a> <a name = "SSEC001.5"></a> <li><code>AbelianGroupsOfExponent(</code><var>n</var><code>) F</code> <p> <a name = "I9"></a> <a name = "I10"></a> returns the class of all abelian groups of exponent dividing <var>n</var>, where <var>n</var> is a positive integer. It is always a subgroup-closed formation. <p> <a name = "SSEC001.6"></a> <li><code>PiGroups(</code><var>pi</var><code>) F</code> <p> <a name = "I11"></a> constructs the class of all <var>pi</var>-groups. <var>pi</var> may be a non-empty class or a set of primes. The result is a subgroup-closed saturated Fitting formation. <p> <a name = "SSEC001.7"></a> <li><code>PGroups(</code><var>p</var><code>) F</code> <p> <a name = "I12"></a> returns the class of all <var>p</var>-groups, where <var>p</var> is a prime. The result is a subgroup-closed saturated Fitting formation. <p> <p> <h2><a name="SECT002">6.2 Pre-defined projector functions</a></h2> <p><p> <a name = "SSEC002.1"></a> <li><code>NilpotentProjector(</code><var>grp</var><code>) A</code> <p> <a name = "I13"></a> groups. For a definition, see <a href="CHAP004.htm#SSEC002.3">Projector</a>. Note that the nilpotent projectors of a finite solvable group equal its a Carter subgroups, that is, its self-normalizing nilpotent subgroups. <p> <a name = "SSEC002.2"></a> <li><code>SupersolvableProjector(</code><var>grp</var><code>) A</code> <p> These functions return a projector for the class of all finite supersolvable groups. For a definition, see <a href="CHAP004.htm#SSEC002.3">Projector</a>. <p> <p> <h2><a name="SECT003">6.3 Pre-defined sets of primes</a></h2> <p><p> <a name = "SSEC003.1"></a> <li><code>AllPrimes V</code> <p> <a name = "I14"></a> installed as value for <code>Characteristic(</code><var>grpclass</var><code>)</code> if the group class <var>grpclass</var> contains cyclic groups of prime order <var>p</var> for arbitrary primes <var>p</var>. <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP005.htm">Previous</a>] [<a href = "theindex.htm">Index</a>] <P> <address>CRISP manual<br>June 2007 </address></body></html>
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