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<html><head><title>[ModIsom] 3 Automorphism groups and Canonical Forms</title></head>
<body text="#000000" bgcolor="#ffffff"> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Nex <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 use 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 # the entries of the canonical table as far as they are bounded gap> C.cano.tab; [ [ <a GF2 vector of length 7>, <a GF2 vector of length 7>, [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ], [ <a GF2 vector of length 7>, <a GF2 vector of length 7>, [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ], [ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ], [ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0, 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ] ], [ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ] ], [ [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), Z(2)^0 ], [ 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2), 0*Z(2) ] ] ] </pre> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Nex <P> <address>ModIsom manual<br>September 2024 </address></body></html> ¤ Dauer der Verarbeitung: 0.28 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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