# The direct extension theorem

J. Group Theory 9 (2006), 307­316 DOI 10.1515/JGT.2006.020 ( de Gruyter 2006 Joseph Ayoub (Communicated by R. M. Guralnick) 1 Introduction Let H and K be finite groups and consider an extension 1 ! H ! G ! K ! 1. In general the isomorphism class of G does not determine the extension. The main result of this paper is it does in the special case that G G H Â K. Our theorem can be restated as follows: Theorem 1.1. Let G ¼ H Â K be a finite group. Suppose that H0 is a normal subgroup of G and assume that H0 G H and G=H0 G G=H. Then H0 is a direct factor of G (i.e. G ¼ H0 Â K0 for some complement K0 ). The corresponding result for finitely generated modules over a noetherian commutative ring is well known (see [2] or [3]). In particular, the result holds for finitely generated abelian groups. 2 Preliminary results In this section, G is any finite group, not necessarily the group that appears in the theorem. 2.1 Subgroups of a direct product G F H D K. We shall often use the following elementary results without http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Group Theory de Gruyter

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