COMPUTATIONAL PROBLEMS IN ABSTRACT ALGEBRA.

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John McKay
In the table, c indicates the cosine of a multiple of 2nlperiod. For example,
2+4c2, occurring in the second row of the table as the character of an
element of period 5 in the fourth conjugacy class, is an abbreviation for
2+4 cos (2X&/5).
REFERENCES
i. J. S. FRAME: The constructive reduction of finite group representations. Proc. Symp.
Pure Maths. (AMS) 6 (1962), 89-99.
2. N. S. MENDELSOHN: Defining relations for subgroups of finite index of groups with a
tinite presentation. These Proceedings, pp. 43-44.
3. J. G. F. FRANCIS: The QR transformation. Pts. 1 & 2. Computer Journal 4 (1961-
1962), 265-271, 332-345.
4. H. S. M. C~XETER~~~ W. 0. J. MOSER: Generators and Relations for Discrete Groups.
Ergebnisse der Mathematik NF 14 (Springer, Berlin 1965).
5. M. HALL and J. K. SENIOR: Groups of Order 2” (n&) (MacMillan, New York, 1964).
6. Z. JANKO: A new finite simple group with Abelian Sylow 2-subgroups and its characterization.
J. of Algebra, 3 (1966), 147-186.
A programme for the calculation of characters
and representations of’nite groups
C. BROTTAND J.NEUB&ER
1. Introdnction. The programme described in this paper is part of a system
of programmes for the investigation of finite groups. Other parts of this
system are described in [3], [4].
,‘I
._
The programme avoids numerical calculations as far as possible.
Instead properties of the given group which are available from other
programmes have been used to construct characters and representations
by the process of induction. Only when this process does not yield all the
required information does the programme use numerical methods.
The programme has been started as a “Diplomarbeit” [l]. We are grateful
to the Deutsche Forschungsgemeinschaft for financial support and to
Prof. K. H. Weise for opportunities given to us at the “Rechenzentrum
der Universitgt Kiel”. We would like to thank Mr. V. Felsch for valuable
-help in connecting this programme with the one described in [3].
1.1. Notations. All groups considered are finite, they are denoted by
CH, . . ..(gl. . . . . gJ is the subgroup generated by the elements gr,
. . ., ge E G; G’ = (x-’ y-lxyl X, y E G) is the commutator subgroup of
G; Cl, . . . . C, are the classes of elements conjugate in G; Cr = {I}, hi
is the number of elements in Ci. The structure constants cijk are defined by
(1.1.1)
Z is the ring of integers, Z, the field of integers module a prime p, C the
complex field, Mr, . . . , M, the set of all absolutely irreducible CG-modules
with dimensions dl, . . . , d,. $J> is the character belonging to Mj, djJ
its value on Ci. All representations considered are C-representations, so
irreducible always means absolutely irreducible.
2. Available programmes for the investigation of finite groups. Our
programme X for the calculation of characters and representations of a
given finite group G makes use of the data determined by a programme @
for the investigation of the lattice of subgroups and of certain other properties
of G. The input for the system of programmes consisting of @ and X
is a set of generators of G given in one of the following ways:
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