COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

7
The construction of the character table of a finite
group from generators and relations
JOHN MCKAY
Introduction. There are six problems in determining the character table
from the generators and defining relations for a finite group. They are
(a) derivation of a faithful representation,
(b) generation of the group elements,
(c) determination of the mapping of an element into its conjugacy class,
(d) derivation of the structure constants of the class algebra,
(e) determination of the numerical values of the characters from the
structure constants, and
(f) derivation of the algebraic from the numerical values.
Use of the methods is illustrated by the construction of the character
table of the simple group JI, of order 175,560, which is given in the
Appendix in the form output by the computer.
G denotes a finite group of order g having r conjugacy classes Ci of
orderh,i = 1, . . . . r. C, is the class inverse to C’i. A(G, C) denotes the
group algebra of G over the complex field C.
Derivation of a faithful representation. Enumeration of the cosets of a
subgroup H of G gives rise to a permutation representation on the generators
and their inverses. The representation so formed is a faithful representation
of the factor group G/N, where N = II x-IHx, known as the “core” of
%EG
H in G. The representation will be a faithful representation of G whenever
H contains no non-trivial normal subgroup of G. There are three requirements
in particular for representations to be useful for computing purposes.
Firstly, the representation of an element should be unique; secondly,
it should be representable within the computer sufficiently economically to
cause no storage problem; and thirdly, it should be such that the product
of two elements can be derived quickly. For the smaller groups these requirements
may be relaxed, but for large groups they are essential.
Both permutation representations and faithful irreducible representations
of minimal degree are suitable for computer work. Multiplication of
permutations is fast but it is often easier to find a matrix representation
1s 89