A characterization of the finite soluble groups
The author proves various interesting results characterizing finite groups which admit certain types of factorizations. In particular, answering a question of Kegel, it is shown (Theorem 1.1) that a finite group in which every maximal subgroup admits an abelian supplement is solvable. Other results include Theorem 1.3: A finite group is solvable if and only if every maximal subgroup \(M\) admits a complement whose commutator subgroup is contained in \(M\). Theorem 1.4 characterizes groups in which every maximal subgroup admits a nilpotent complement (if and only if \(G/O_\infty(G)=1\) or \(L_2(7)\)). These are precisely the groups in which every maximal subgroup has prime power index [see \textit{R. Guralnick}, J. Algebra 81, 304-311 (1983; Zbl 0515.20011)]. Theorem 1.5 characterizes those groups in which every maximal subgroup admits a solvable complement (the simple groups occurring are \(L_2(q)\) with \(q=5,7,9,11\)). The key idea is to reduce to studying simple groups and to use the classification of simple groups and the classification of (maximal) factorizations of almost simple groups by \textit{M. W. Liebeck, C. E. Praeger}, and \textit{J. Saxl} [Mem. Am. Math. Soc. 432 (1990; Zbl 0703.20021)].
scientific article; zbMATH DE number 1295488