<html><head><title>[CRISP] 7 Lists of normal subgroups</title></head>
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<h1>7 Lists of normal subgroups</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP007.htm#SECT001">Functions for normal and characteristic subgroups</a>
<li> <A HREF="CHAP007.htm#SECT002">Functions for the socle of finite groups</a>
</ol><p>
<p>
The algorithms in <font face="Gill Sans,Helvetica,Arial">CRISP</font> can also be used to compute certain normal subgroups of a finite soluble
group efficiently. In particular, <font face="Gill Sans,Helvetica,Arial">CRISP</font> provides fast methods for computing all normal subgroups, all minimal normal subgroups, and the socle of a finite soluble group.
<p>
<p>
<h2><a name="SECT001">7.1 Functions for normal and characteristic subgroups</a></h2>
<p><p>
<a name = "SSEC001.1"></a>
<li><code>NormalSubgroups(</code><var>grp</var><code>) A</code>
<p>
For finite soluble groups <var>grp</var>, <font face="Gill Sans,Helvetica,Arial">CRISP</font> provides an efficient method to compute <code>NormalSubgroups</code> (see <a href="../../../doc/ref/chap39.html#X80237A847E24E6CF">NormalSubgroups</a>).
<p>
<a name = "SSEC001.2"></a>
<li><code>CharacteristicSubgroups(</code><var>grp</var><code>) A</code>
<p>
returns a list containing all characteristic subgroups of the finite soluble group <var>grp</var>.
<code>CharacteristicSubgroups</code> calls <code>AllInvSgrsWithQPropUnderAction</code>.
<p>
<a name = "SSEC001.3"></a>
<li><code>MinimalNormalSubgroups(</code><var>grp</var><code>) A</code>
<p>
<a name = "I0"></a>
<font face="Gill Sans,Helvetica,Arial">CRISP</font> provides an efficient method to compute a list of all minimal normal subgroups of <var>grp</var> (see <a href="../../../doc/ref/chap39.html#X86FDD9BA819F5644">MinimalNormalSubgroups</a>).
<p>
<a name = "SSEC001.4"></a>
<li><code>MinimalNormalPSubgroups(</code><var>grp</var><code>, </code><var>p</var><code>) A</code>
<p>
<a name = "I1"></a>
For a prime <var>p</var>, this function computes a list of all <var>p</var>-subgroups which are minimal among the nontrivial
normal subgroups of <var>grp</var>.
<p>
<a name = "SSEC001.5"></a>
<li><code>AbelianMinimalNormalSubgroups(</code><var>grp</var><code>) A</code>
<p>
This computes a list of all minimal normal subgroups of <var>grp</var> which are abelian. If <var>grp</var> is soluble, this list coincides with the list of all
minimal normal subgroups of <var>grp</var>.
<p>
<p>
<h2><a name="SECT002">7.2 Functions for the socle of finite groups</a></h2>
<p><p>
<a name = "SSEC002.1"></a>
<li><code>Socle(</code><var>grp</var><code>) A</code>
<p>
<font face="Gill Sans,Helvetica,Arial">CRISP</font> provides a method for <code>Socle</code> (see <a href="../../../doc/ref/chap39.html#X81F647FA83D8854F">Socle</a>) for which works for
all finite soluble groups <var>grp</var>. The socle of a group <var>grp</var> is the subgroup
generated by all minimal normal subgroups of <var>grp</var>. See also <a href="CHAP007.htm#SSEC002.2">SolubleSocle</a> and
<a href="CHAP007.htm#SSEC002.5">PSocle</a> below.
<p>
<pre>
gap> Size(Socle( DirectProduct(DihedralGroup(8), CyclicGroup(12))));
12
</pre>
<p>
<a name = "SSEC002.2"></a>
<li><code>AbelianSocle(</code><var>grp</var><code>) A</code>
<a name = "SSEC002.2"></a>
<li><code>SolubleSocle(</code><var>grp</var><code>) A</code>
<a name = "SSEC002.2"></a>
<li><code>SolvableSocle(</code><var>grp</var><code>) A</code>
<p>
This function computes the soluble socle of <var>grp</var>. The soluble socle of a group <var>grp</var> is the
subgroup generated by all minimal normal soluble subgroups of <var>grp</var>.
<p>
<a name = "SSEC002.3"></a>
<li><code>SocleComponents(</code><var>grp</var><code>) A</code>
<p>
This function returns a list of minimal normal subgroups of <var>grp</var> such
that the socle of <var>grp</var> (see <a href="CHAP007.htm#SSEC002.1">Socle</a>) is the direct product of these minimal normal
subgroups. Note that, in general, this decomposition is not unique. Currently,
this function is only implemented for finite soluble groups. See also
<a href="CHAP007.htm#SSEC002.4">SolubleSocleComponents</a> and <a href="CHAP007.htm#SSEC002.6">PSocleComponents</a>.
<p>
<a name = "SSEC002.4"></a>
<li><code>AbelianSocleComponents(</code><var>grp</var><code>) A</code>
<a name = "SSEC002.4"></a>
<li><code>SolubleSocleComponents(</code><var>grp</var><code>) A</code>
<a name = "SSEC002.4"></a>
<li><code>SolvableSocleComponents(</code><var>grp</var><code>) A</code>
<p>
This function returns a list of soluble minimal normal subgroups of <var>grp</var> such
that the socle of <var>grp</var> (see <a href="CHAP007.htm#SSEC002.1">Socle</a>) is the direct product of these minimal normal
subgroups. Note that, in general, this decomposition is not unique.
<p>
<a name = "SSEC002.5"></a>
<li><code>PSocle(</code><var>grp</var><code>, </code><var>p</var><code>) A</code>
<p>
If <var>p</var> is a prime, the <var>p</var>-socle of a group <var>grp</var> is the subgroup
generated by all minimal normal <var>p</var>-subgroups of <var>grp</var>.
<p>
<a name = "SSEC002.6"></a>
<li><code>PSocleComponents(</code><var>grp</var><code>, </code><var>p</var><code>) A</code>
<p>
For a prime <var>p</var>, this function returns a list of minimal normal <var>p</var>-subgroups of <var>grp</var>
such that the <var>p</var>-socle of <var>grp</var> (see <a href="CHAP007.htm#SSEC002.5">PSocle</a>) is the direct product of these minimal normal
subgroups. Note that, in general, this decomposition is not unique.
<p>
<a name = "SSEC002.7"></a>
<li><code>PSocleSeries(</code><var>grp</var><code>, </code><var>p</var><code>) A</code>
<p>
For a prime <var>p</var>, this function returns an ascending <var>grp</var>-composition series of the <var>p</var>-socle of <var>grp</var>.
<p>
<p>
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<address>CRISP manual<br>August 2025
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