COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

On a programme for the determination of the
automorphism group of a finite group
V. FELSCH and J. NEUB~SER
THE programme A for the determination of the automorphism group A(G)
of a finite group G is part of a system of programmes for the investigation
of finite groups implemented on an Electrologica Xl at the “Rechenzentrum
der Universitat Kiel”. A detailed description [3] of A has been published
in Numerische Mathematik. Therefore here we give only a short summary.
Notations are as in [l].
The programme A makes use of information about the lattice of subgroups
of the group G, provided by a programme Cl) described in [2]. The
programme A works as follows.
1. A system of generators and defining relations of G is determined. It is
used later to decide whether a mapping from G onto G is a homomorphism.
There are three cases:
1.1. If G is soluble, a system of generators a0 = 1, al, . . ., a, is chosen
such that the subgroups Ui = ( ao, . . . , ai) form a subnormal series.
For i = l( l)r, let ai be the least positive integer with a? E Vi-l. Then
and
a? = 1, a;’ = ay,l . . . ayL%l, i = 2(l)r,
a,@i = @‘i’1 . . . &?ikml ak, l=si-=k=sr,
are defining relations of G.
1.2. If G is nonsoluble, A searches for generators a0 = 1, al,. . . , a, of
G with the property: For i = l(l)r there exist integers ai>l such that each
gEUi=(al, . . . . ai ) is obtained exactly once as g = &rl. . . a;* with
0 =S &j