Abstract
We will prove an analogue of Glauberman’s ZJ-Theorem, [4] and [6, Theorem 8.2.1, p. 279], that can be used to study finite groups that admit a coprime group of automorphisms. This analogue is unusual in that no hypothesis of p-stability is required. Used in conjunction with the Bender Method, it may make it possible to prove very general results about finite groups that admit a coprime group of automorphisms. The reader is referred to [7] and [8] for a fuller discussion of the Bender Method, Glauberman’s ZJ-Theorem and p-stability. Before stating the main result of this paper, we introduce some notation. If R and G are groups then we say that R acts coprimely on G if R acts as a group of automorphisms on G, if R and G have coprime orders and if at least one of R or G is soluble. Suppose that R acts coprimely on G and that p is a prime. Define
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