On Sylow p-subgroups of the general linear group over commutative local ring of characteristic p^s
by
Alexander Tilischak
Uzhgorod University, Uzhgorod, Ukraine

Suprunenko [1] has proved, that any Sylow p-subgroup of the general linear group GL(n, F) over a field F of characteristic p is conjugated to unitriangular subgroup UT(n, F) of this group. Gudivok and Rud'ko [2] have shown, that Sylow p-subgroups of the general linear group GL(n, L) of the degree n > 1 over noetherian local integral domain L of characteristic p are pairly conjugated if and only if L is a principal ideal domain.

We are investigating when the Sylow p-subgroups of the general linear group over a commutative local ring of characteristic p^s are pairly conjugated. We shall call a domain K with identity by the Bezu ring if any finitely generated ideal of the ring K is a principal ideal. rad K will be denotes a prime radical of the ring K. Using some results of [3] of representation theory of finite p-groups over rings makes it possible to prove the next theorem.

Theorem. Let K be a commutative local ring with identity of characteristic p^s (s > 0, p is a prime), n is a natural (n > 1). Sylow p-subgroups of the group GL(n, K) are pairly conjugated if and only if K/rad K is a Bezu ring.

References

[1] D. A. Suprunenko, Linear p-groups, Dokl. Akad. Nauk BSSR 4 (1960), 233-235.

[2] P. M. Gudivok, V. P. Rud'ko, On Sylow subgroups of the general linear group over integral domains, Dopovidi NAN Ukrayiny 8 (1995), 5-7.

[3] P. M. Gudivok, A. A. Tilischak, On irreducible modular representation of finite p-group over commutative local rings, Nauk. visnik Uzhgorod. univ., ser. matem. 3 (1998), 78-83.

Date received: July 18, 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-38.