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The maximal factorizations of the finite simple groups and their automorphism groups. (English) Zbl 0703.20021
Mem. Am. Math. Soc. 432, 151 p. (1990).
In this long but clearly written paper, the authors determine all maximal factorizations of finite groups G which satisfy \(L\trianglelefteq G\leq Aut(L)\), for some non-abelian simple group L. That is, all pairs (A,B) of maximal subgroups of G not containing L, with \(G=AB\). The major part of the work is involved with the case L classical. Complete proofs are also given for the cases L alternating and sporadic, while the exceptional case is merely sketched: a complete proof is given by C. Hering, M. Liebeck and J. Saxl [J. Algebra 106, 517-527 (1987; Zbl 0607.20012)].
The proof is based on the idea that at least one of A and B must be large, and therefore known. It then proceeds case by case to see how the (suitably defined) “large” prime divisors of the order of G can be distributed between A and B.
This work has an important application to the theory of maximal subgroups of \(A_ n\) and \(S_ n:\) this result and a sketch of the proof are given in Chapter 9, while complete proofs, due to the authors can be found in [J. Algebra 111, 365-383 (1987; Zbl 0632.20011)]. The latter result is usually expressed as a “classification of the maximal subgroups of \(A_ n\) and \(S_ n''\), but in fact it lists those candidates for maximal subgroups (namely, primitive almost simple groups) which are not maximal. Producing the list of candidates is an impossible problem which is left to the reader!
For a more detailed summary of the problem, results, methods and applications, Chapter 1 of the present paper is highly recommended.
Reviewer: R.A.Wilson

20D40 Products of subgroups of abstract finite groups
20B35 Subgroups of symmetric groups
20D06 Simple groups: alternating groups and groups of Lie type
20G40 Linear algebraic groups over finite fields
20D08 Simple groups: sporadic groups
20E28 Maximal subgroups
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