<html><head><title>[ModIsom] 3 Automorphism groups and Canonical Forms</title></head>
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<h1>3 Automorphism groups and Canonical Forms</h1><p>
<P>
<H3>Sections</H3>
<oL>
<li> <A HREF="CHAP003.htm#SECT001">Automorphism groups</a>
<li> <A HREF="CHAP003.htm#SECT002">Canonical forms</a>
<li> <A HREF="CHAP003.htm#SECT003">Example of canonical form computation</a>
</ol><p>
<p>
We refer to <a href="biblio.htm#Eic07"><cite>Eic07</cite></a> for background on the algorithms used in this
Chapter. Throughout the chapter, we assume that <i>F</i> is a finite field.
<p>
<p>
<h2><a name="SECT001">3.1 Automorphism groups</a></h2>
<p><p>
Let <i>T</i> be a nilpotent table over <i>F</i>. The following function can be used
to determine the automorphism group of the algebra described by <i>T</i>. The
automorphism group is determined as subgroup of <i>GL</i>(<i>T</i>.<i>dim</i>, <i>T</i>.<i>fld</i>) given
by generators and its order. There is a variation available to determine
the automorphism group of a modular group algebra <i>FG</i>, where <i>F</i> is a finite
field and <i>G</i> is a <i>p</i>-group.
<p>
<a name = "SSEC001.1"></a>
<li><code>AutGroupOfTable( T ) F</code>
<a name = "SSEC001.1"></a>
<li><code>AutGroupOfRad( FG ) F</code>
<p>
In both cases, the automorphism group is described by a record. The
matrices in the lists <i>glAutos</i> and <i>agAutos</i> generate together the
automorphism group. The matrices in <i>agAutos</i> generate a <i>p</i>-group.
The entry <i>size</i> contains the order of the automorphism group.
<p>
<p>
<h2><a name="SECT002">3.2 Canonical forms</a></h2>
<p><p>
Let <i>T</i> be a nilpotent table. The following function can be used to determine
the automorphism group of <i>T</i> if the underlying field of <i>T</i> is finite. The
canonical form is a nilpotent table which is unique for the isomorphism type
of the algebra defined by <i>T</i>. Again there a variation available for modular
group algebras.
<p>
<a name = "SSEC002.1"></a>
<li><code>CanonicalFormOfTable( T ) F</code>
<a name = "SSEC002.1"></a>
<li><code>CanonicalFormOfRad( FG ) F</code>
<p>
The automorphism group of <i>T</i> is determined as a side-product of computing
the canonical form. The following functions can be used to return both.
<p>
<a name = "SSEC002.2"></a>
<li><code>CanoFormWithAutGroupOfTable( T ) F</code>
<a name = "SSEC002.2"></a>
<li><code>CanoFormWithAutGroupOfRad( FG ) F</code>
<p>
In both cases, these functions return a record with entries <i>cano</i> and
<i>auto</i>.
<p>
<p>
<h2><a name="SECT003">3.3 Example of canonical form computation</a></h2>
<p><p>
We compute the automorphism group and a canonical form for the
modular group algebra of the dihedral group of order 8.
<p>
<pre>
gap> A := GroupRing(GF(2), SmallGroup(8,3));;
gap> T := TableByWeightedBasisOfRad(A);;
gap> C := CanoFormWithAutGroupOfTable(T);;
# check that the canonical form is not equal to T
gap> CompareTables(C.cano, T);
false
# the order of the automorphism group
gap> C.auto.size;
512
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