Finite rational matrix groups

W. Plesken and G. Nebe

I. Introduction

This paper continues the investigation and classification of the maximal finite sub-

groups of GLn(Q) started in [Pie 91]. There the study of these groups was essentially

reduced to that of irreducible maximal finite subgroups of GLn(Q), henceforth ab-

breviated as r.i.m.f. groups, the r.i.m.f. groups were determined up to degree 10,

the imprimitive r.i.m.f. groups were discussed, and the automorphism groups of most

irreducible root systems were exhibited as important examples. Here we exhibit fur-

ther infinite series of r.i.m.f. groups, namely the ones of type L2(p) of degree p ± 1

for odd prime numbers p, cf. Chapter V. In particular, all r.i.m.f. groups of degree

p — 1, whose order is divisible by p will be classified. Secondly all r.i.m.f. groups

up to degree n 23, n ^ 16 will be determined. Actually, the more elaborate task

of classifying the irreducible maximal finite subgroups of GLn(2£) has already been

performed in [Pie 85] for prime dimensions up to 23, so that we need not deal with

that case anymore. The various lattices in Q x n invariant under the action of a r.i.m.f

group G, define embeddings of G into GLn{!E). Therefore our results will also give

the irreducible maximal finite subgroups of GLn(Z), which stay maximal finite inside

GLn(Q). If viewed as lattices with a G-invariant quadratic form these lattices have

nice arithmetic and geometric properties and are of interest in themselves. It might

happen that some irreducible rational matrix group acts on the lattices of two or more

r.i.m.f groups. This gives interrelation between our r.i.m.f. groups, which are encoded

in the complex Mlnrr(Q) defined in [Pie 91]:

Received by the editor November 13, 1991, and in revised form November 6, 1992.

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