COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

10 J. Neubiiser
nilpotent, supersoluble, soluble or perfect and whether it is normal, subnormal
or self-normalizing in G. The order, normalizer, centralizer and
classes of conjugate elements of U are found. Also certain characteristic
subgroups such as the upper and lower central series, commutator series
Frattini and Fitting subgroups are determined.
3.2.6. Two sets of programmes developed by R. Yates [Ya l] and R.
Segovia and H. V. McIntosh [Se l] determine the lattice of subgroups
and some of these structural details for groups of special types.
3.2.7. Some more extensive applications of such programmes have been
made. R. Biilow and J. Neubtiser [Bii 1,2] determined all integral classes of
4X4 integral matrix groups by analysing the 9 classes of maximal 4 X 4
integral groups, found by E. C. Dade [Da 11. A catalogue of the lattices of
subgroups, etc., of the groups of order 4 100, omitting orders 64 and 96,
has been compiled by J. Neubtiser [Ne 31 using the Kiel programmes.
3.3. Automorphism groups. W. D. Maurer’s system of programmes
[Ma 8] also contains a programme which checks for the existence of an
isomorphism between two given groups by constructing partial isomorphisms
in a tree-of-trial-and-error procedure. Programmes for detecting
isomorphisms of graphs [Be 2; Su l] may also be used in such a way.
The only programme known to me [Fe 5,7, 8] for the determination of
the automorphism group A(G) of a finite group G uses the structural data
obtained by the subgroup lattice programme [Fe 5,6]. The basic idea is that
an automorphism induces a lattice automorphism on the lattice of subgroups
which preserves order, conjugacy, centralizers, etc. If a subgroup U
is chosen it is usually rather easy by visual inspection of a diagram representing
the lattice of subgroups to find all candidates for images of U under
automorphisms of G. This vague idea has been made precise by introducing
and studying certain equivalence relations on the set of all subgroups which
are then used in the construction of A(G).
For a soluble group which is not of prime power order L. Gerhards
[Ge 2] gives an outline of a programme based on P. Hall’s results [Ha 8]
about Sylow systems. In this programme (which is not yet implemented)
the automorphism group of G will be constructed from “extensible” automorphisms
of the Sylow subgroups in a Sylow system and inner automorphisms.
3.4. Characters and representations.
3.4.1. There are a number of programmes for the determination of the
absolutely irreducible characters of a finite group G. Most of these start
from the class multiplication coefficients and lead to some kind of numerical
computation. For the theoretical background to this procedure see e.g.
[Cu 1].
Let Cl, . . . . C, be the classes of conjugate elements in G, Cr = {l}, hi
the number of elements in Cj, ~1, . . . , x’ the absolutely irreducible charac-
Investigations of groups on computers 11
ters, 4 the dimension of $, and xj the value of $for an element of Ci. The
class multiplication coefficients are defined by
CiCj = i CijkCke
k=l
Defining ti = 2 fi one has n+$ = i c&kw,$.
k=l
Henceforeachs, l&s&r,
the r values n$, l’=~ 24 r, belonging to the sth character satisfy the r2 equations
$pjk- dikxj)xk = 0 for 1 < i, j s r. (7
Considering for each jO, l== j,, s r, in turn the r equations obtained by fixing
j = je, one sees that the vector ws = (tir, . . . , 4) is an eigenvector of
the matrix (Cam) belonging to the eigenvalue $0, and further that these
vectors Ws, 1 d s 4 r, are (up to factors) the only common eigenvectors of all
these matrices (c&, l== jo < r.
3.4.2. [Cu 11: If for a certain jO the matrix (cii&) has r different eigenvalues,
the eigenvector spaces are one-dimensional and hence yield uniquely
(up to scalar multiples) the vectors w’, 14 s 4 r. These factors are deterhl
mined from tir = -d xi = 1. From the orthogonality relations of the characters
one has
s
df i$l d$ = [Cl.
From this dl, . . . , d, are found and then the characters. As there are simple
examples in which none of the matrices (Cij~) has r distinct eigenvalues,
this method is not always applicable.
Essentially this method is applied for all matrices (cliok) in a programme
by S. Flodmark and E. Blokker [F13] to obtain irreducible characters. These
are then used to reduce the regular representation into representations
El, . ..,Erofdegreedf, . . . . d,” respectively. Ej contains the 4 copies of the
jth absolutely irreducible representation that are contained in the regular
representation. For further reduction of the Ej into irreducible representations,
according to a private communication of S. Flodmark, a “numerical
iteration procedure” is used. The author regrets that he cannot supply
details about this, as he heard about the programme only when this article
had already gone to press.
3.4.3. For hand computation W. Burnside [Bu l] gives the following
method: Multiplying the equations (*) with a tied j by an indeterminate uj
and summing over all j yields