The direct extension theorem Joseph Ayoub Summary: The direct extension theorem Joseph Ayoub 25th March 2004 Abstract The problem of group extension can be divided into two sub-problems. 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", Anne-Marie Aubert as well as Charles-Antoine 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 Collections: Mathematics