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\Chapter{Automorphism groups and Canonical Forms}
We refer to
\cite{Eic07} for background on the algorithms used in this
Chapter. Throughout the chapter, we assume that $F$ is a finite field.
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\Section{Automorphism groups}
Let $T$ be a nilpotent table over $F$. The following function can be used
to determine the automorphism group of the algebra described by $T$. The
automorphism group is determined as subgroup of $GL(T
\.dim, T
\.fld)$ given
by generators and its order. There is a variation available to determine
the automorphism group of a modular group algebra $FG$, where $F$ is a finite
field and $G$ is a $p$-group.
\> AutGroupOfTable( T ) F
\> AutGroupOfRad( FG ) F
In both cases, the automorphism group is described by a record. The
matrices in the lists $glAutos$ and $agAutos$ generate together the
automorphism group. The matrices in $agAutos$ generate a $p$-group.
The entry $size$ contains the order of the automorphism group.
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\Section{Canonical forms}
Let $T$ be a nilpotent table. The following function can be used to determine
the automorphism group of $T$ if the underlying field of $T$ is finite. The
canonical form is a nilpotent table which is unique for the isomorphism type
of the algebra defined by $T$. Again there a variation available for modular
group algebras.
\> CanonicalFormOfTable( T ) F
\> CanonicalFormOfRad( FG ) F
The automorphism group of $T$ is determined as a side-product of computing
the canonical form. The following functions can be used to return both.
\> CanoFormWithAutGroupOfTable( T ) F
\> CanoFormWithAutGroupOfRad( FG ) F
In both cases, these functions return a record with entries $cano$ and
$auto$.
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\Section{Example of canonical form computation}
We compute the automorphism group and a canonical form for the
modular group algebra of the dihedral group of order 8.
\beginexample
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) ] ] ]
\endexample
