 
Summary: The direct extension theorem
Joseph Ayoub
25th March 2004
Abstract
The problem of group extension can be divided into two subproblems. The first is to find all the
possible extensions of H by K. The second is to find the different ways a group G can arise as an
extension of H by K. Here we prove that the direct product H × K can arise as an extension of H by
K in an essentially unique way: that is the direct extension. I would like to thank Yacine Dolivet for
drawing my attention to the direct "extension theorem", AnneMarie Aubert as well as CharlesAntoine
Louet for their support and Robert Guralnick for suggesting me better proofs of propositions 2.3 and 3.1
Contents
1 Statement of the theorem 2
2 A few preliminary general results 2
2.1 Subgroups of a direct product G = H.K . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Coprime direct factors of a finite group G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.3 Strongly decomposable subgroups of G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 Two special cases of the theorem 3
3.1 The case of commutative groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3.2 The case where G = G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
4 A few preliminary lemmas 4
