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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

  

Source: Ayoub, Joseph - Institut für Mathematik, Universität Zürich

 

Collections: Mathematics