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<html><head><title>[AutPGrp] 3 The underlying function</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 The underlying function</h1><p> <p> Underlying the method installation for <code>AutomorphismGroup</code> is the function <code>AutomorphismGroupPGroup</code>. This function is intended for expert users who wish to influence the steps of the algorithm. Note also that <code>AutomorphismGroup</code> will always choose default values. <p> <a name = ""></a> <li><code>AutomorphismGroupPGroup( </code><var>G</var><code> [,</code><var>flag</var><code>] ) F</code> <p> The input is a finite <i>p</i>-group as above and an optional <var>flag</var> which can be true or false. Here the filters for <var>G</var> need not be set, but they should be true for <var>G</var>. The possible values for <var>flag</var> are considered later in Chapter <a href="CHAP004.htm">Influencing the algorithm</a>. If <var>flag</var> is not supplied, the algorithm proceeds similarly to the method installed for <code>AutomorphismGroup</code>, but it produces slightly more detailed output. The output of the function is a record which contains the following fields: <p> <p> <dl compact> <dt><code>glAutos</code> <dd> a set of automorphisms which together with <code>agAutos</code> generate the automorphism group; <p> <dt><code>glOrder</code> <dd> an integer whose product with the <code>agOrders</code> gives the size of the automorphism group; <p> <dt><code>agAutos</code> <dd> a polycyclic generating sequence for a soluble normal subgroup of the automorphism group; <p> <dt><code>agOrder</code> <dd> the relative orders corresponding to <code>agAutos</code>; <p> <dt><code>one</code> <dd> the identity element of the automorphism group; <p> <dt><code>group</code> <dd> the underlying group <var>G</var>; <p> <dt><code>size</code> <dd> the size of the automorphism group. </dl> <p> We do not return an automorphism group in the standard form because we wish to distinguish between <code>agAutos</code> and <code>glAutos</code>; the latter act non-trivially on the Frattini quotient of <var>G</var>. This hybrid-group description of the automorphism group permits more efficient computations with it. The following function converts the output of <code>AutomorphismGroupPGroup</code> to the output of <code>AutomorphismGroup</code>. <p> <a name = ""></a> <li><code>ConvertHybridAutGroup( </code><var>A</var><code> ) F</code> <p> <pre> gap> LoadPackage("autpgrp", false); true gap> H := SmallGroup (729, 34); <pc group of size 729 with 6 generators> gap> A := AutomorphismGroupPGroup(H); rec( glAutos := [ ], glOrder := 1, agAutos := [ Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1^2, f2, f3^2*f4, f4, f5^2*f6, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f2^2, f1, f3*f5^2, f5^2, f4*f6^2, f6^2 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1^2, f2^2, f3*f4^2*f5^2*f6, f4^2*f6, f5^2*f6, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f3, f2, f3*f5^2, f4*f6^2, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f3, f3*f4, f4, f5*f6, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f4, f2, f3*f6^2, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f4, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f5, f2, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f5, f3*f6, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1*f6, f2, f3, f4, f5, f6 ], Pcgs([ f1, f2, f3, f4, f5, f6 ]) -> [ f1, f2*f6, f3, f4, f5, f6 ] ], agOrder := [ 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3 ], one := IdentityMapping( <pc group of size 729 with 6 generators> ), group := <pc group of size 729 with 6 generators>, size := 52488 ) gap> ConvertHybridAutGroup( A ); <group of size 52488 with 11 generators> </pre> <p> Let <var>A</var> be the automorphism group of a <i>p</i>-group <i>G</i> as computed by <code>AutomorphismGroupPGroup</code>. Then the following function can compute a pc group isomorphic to the solvable part of <var>A</var> stored in the record component <var>A</var>.agGroup. This solvable part forms a subgroup of the automorphism group which contains at least the automorphisms centralizing the Frattini factor of <i>G</i>. The pc group facilitates various further computations with <var>A</var>. <p> <a name = ""></a> <li><code>PcGroupAutPGroup( </code><var>A</var><code> ) F</code> <p> computes a pc presentation for the solvable part of the automorphism group <var>A</var> defined by <var>A</var>.agGroup. <var>A</var> is the output of the function <code>AutomorphismGroupPGroup</code>. <p> <pre> gap> H := SmallGroup (729, 34);; gap> A := AutomorphismGroupPGroup(H);; gap> B := PcGroupAutPGroup( A ); <pc group of size 52488 with 11 generators> gap> I := InnerAutGroupPGroup( B ); Group([ f5, f4^2*f8, f6^2*f9^2, f11^2, f10^2, <identity> of ... ]) </pre> <p> <p> [<a href = "chapters.htm">Up</a>] [<a href ="CHAP002.htm">Previous</a>] [<a href ="CHAP004.htm">Nex <P> <address>AutPGrp manual<br>April 2025 </address></body></html> ¤ Dauer der Verarbeitung: 0.24 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28