On linear groups of small degrees over residue rings
by
Anatolii S. Kondratiev
Institute of Mathematics and Mechanics UB RAS, Ekaterinburg, Russia
Coauthors: Alexander E. Zalesskii

A new aspect of the classical problem of determining finite linear groups of small degrees over fields gives the following problem.

Problem. Describe finite linear groups of small degrees over local residue rings of Z.

The motivation is that linear groups over such rings are exactly automorphism groups of finite homocyclic p-groups. Let p be a prime and k be a positive integer. Let R_k denote the residue ring Z/p^kZ of the ring Z modulo p^k. Set H_k=GL(n, R_k). The group H_k is an extention of O_p(H_k) by GL(n, p). Unlike to the case k=1, for k > 1 very little is known about subgroups of H_k=GL(n, R_k) other than normal ones. In general, H_k is not splittable over O_p(H_k) for k > 1, i.e., H_k does not have a subgroup isomorphic to GL(n, p). This means that the knowledge of subgroups of GL(n, p) does not lead to the knowledge of subgroups of H_k for k > 1. In this connection, the problem arise: to determine liftable subgroups of GL(n, p) where a subgroup G of GL(n, p) is called liftable to H_k if the preimage of G in H_k is a split extension of O_p(H_k) by G.

In the talk we will discuss a classification of the liftable absolutely irreducible quasi-simple subgroups G of GL(n, p) for p > 3 and n < 28. In the proofs we essentially use the classification of absolutely irreducible finite quasisimple subgroups of GL(n, p) for n < 28 announced in [1].

References

[1] A. S. Kondratiev, Finite linear groups of small degree, In: The Atlas of finite groups: Ten years on, London Math. Soc. Lect. Note Ser., 249, Cambridge University Press, Cambridge, 1998, pp. 139-148.

Date received: July 14, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafe-33.