JOURNAL OF ALGEBRA 22, 137-l 60 (1972) Faithful Representations of p Groups at Characteristic p, II G. J. JANUSZ Department of Mathematics, University of Illinois, Urbana, Illinois 61801 Communicated by W. Feit Received June 22, 1971 INTRODUCTION In this paper we continue to study representations of p groups over fields of characteristic p as begun in [7]. Different questions will be considered here and the results in [7] are needed only in Section 2. This work is motivated in part by two results in the literature. The first is the theorem of D. G. Higman [4] which says that a p group must be cyclic if it has only a finite number of indecomposable representations at characteristic p. The second is the determination of all the indecomposable representations of the noncyclic group of order four by Bashev [I], Conlon [2], and also Heller and Reiner [5]. Of particular interest is the result that there exist just two inequivalent representations of this group in any odd dimension greater than or equal to three over any field of characteristic two. This result is also proved by Johnson [9]. The main result of section one shows these groups are distinguished among abelian groups by these properties. In particular, we show that a noncyclic abelian p group of order not equal to four has infinitely many indecomposable representations in each dimension d if d > 2 and if the ground field is infinite. Moreover, if p(d - 1) is greater than or equal to the exponent of G there exist infinitely many indecomposable representations in dimension d which represent the group faithfully. In Section 2 we consider a non-abelian p group G and an infinite field K. We show there exist infinitely many inequivalent indecomposable K(G)- modules M with the property that M represents G faithfully but no proper submodule or proper homomorphic image of M is faithful for G. This may be somewhat unexpected in view of the results in [7] which show that such modules are isomorphic to principal left ideals in K(G). It would be possible to use modules constructed in this section along with the sort of construction appearing in section one to show that G has infinitely many indecomposable representations in infinitely many different dimensions. 137 Copyright 0 1972 by Academic Press, Inc. All rights of reproduction in any form reserved. brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Elsevier - Publisher Connector