On classifying all full factorisations and multiple-factorisations of the finite almost simple groups
A finite group \(G\) is called almost simple, if \(T\leq G\leq\Aut(T)\) for some simple, non-Abelian group \(T\). The paper under review is concerned with factorizations \(G=AB\) of almost simple groups, where \(A\), \(B\) are proper subgroups of \(G\). Firstly, so-called full factorizations are determined. In this case \(T\not\leq A\), \(T\not\leq B\) and \(\pi(T)\subseteq\pi(A)\cap\pi(B)\) where the function \(\pi\) denotes the set of prime divisors of the order of a finite group. Secondly, multiple factorizations \(\mathcal M\) of almost simple groups are considered. Here \(\mathcal M\) denotes a set of subgroups of \(G\) with \(|{\mathcal M}|\geq 2\) and \(T\not\leq A\), \(T\not\leq B\), \(G=AB\) for all pairs \(A\), \(B\) in \(\mathcal M\). The authors determine all multiple factorizations in the following cases: (a) \(\mathcal M\) is maximal, i.e. all \(A\) in \(\mathcal M\) are maximal subgroups. (b) \(\mathcal M\) is strong, i.e. \(G=A(B\cap C)\) for all triples \(A\), \(B\), \(C\) in \(\mathcal M\). These factorizations are needed in the investigation of quasiprimitive groups by the authors [Tech. Rep. No. 1996/14, Dept. Math. Comput. Sci., Univ. Leicester (1996)]. The paper is based on the classification of maximal factorizations of almost simple groups by \textit{M. W. Liebeck}, \textit{C. E. Praeger} and \textit{J. Saxl} [Mem. Am. Math. Soc. 432 (1990; Zbl 0703.20021)].
scientific article; zbMATH DE number 1218692
scientific article