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%A underl.tex AutPGrp documentation Bettina Eick
%A Eamonn O'Brien
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\Chapter{The underlying function}
Underlying the method installation for `AutomorphismGroup
'
is the function `AutomorphismGroupPGroup
'. This function is
intended for expert users who wish to influence the steps of
the algorithm. Note also that `AutomorphismGroup
' will always
choose default values.
\> AutomorphismGroupPGroup( <G> [,<flag>] ) F
The
input is a finite $p$-group as above and an optional <flag>
which can be true or false. Here the filters for <G> need not be
set, but they should be true for <G>. The possible values for <flag>
are considered later in Chapter
"Influencing the algorithm". If
<flag> is not supplied, the algorithm proceeds similarly to the
method installed for `AutomorphismGroup
', but it produces slightly
more detailed output. The output of the function is a record
which contains the following fields:
\beginitems
`glAutos
' & a set of automorphisms which together with `agAutos'
generate the automorphism group;
`glOrder
' & an integer whose product with the `agOrders' gives
the size of the automorphism group;
`agAutos
' & a polycyclic generating sequence for a soluble normal
subgroup of the automorphism group;
`agOrder
' & the relative orders corresponding to `agAutos';
`one
' & the identity element of the automorphism group;
`group
' & the underlying group ;
`size
' & the size of the automorphism group.
\enditems
We do not return an automorphism group in the standard form
because we wish to distinguish between `agAutos
' and `glAutos';
the latter act non-trivially on the Frattini quotient of <G>. This
hybrid-group description of the automorphism group permits more
efficient computations with it. The following function converts
the output of `AutomorphismGroupPGroup
' to the output of
`AutomorphismGroup
'.
\> ConvertHybridAutGroup( <A> ) F
\beginexample
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>
\endexample
Let <A> be the automorphism group of a $p$-group $G$ as computed by
`AutomorphismGroupPGroup
'. Then the following function can compute
a pc group isomorphic to the solvable part of <A> stored in the record
component <A>.agGroup. This solvable part forms a subgroup of the
automorphism group which contains at least the automorphisms centralizing
the Frattini factor of $G$. The pc group facilitates various further
computations with <A>.
\> PcGroupAutPGroup( <A> ) F
computes a pc presentation for the solvable part of the automorphism
group <A> defined by <A>.agGroup. <A> is the output of the function
`AutomorphismGroupPGroup
'.
\beginexample
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 ... ])
\endexample
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