Products of groups and group classes
This paper mainly concerns finite triply factorized groups \(G = AB = AC = BC\), where \(A\), \(B\) and \(C\) are subgroups. Generalizing previous results of \textit{O. H. Kegel} [Math. Z. 87, 42-48 (1965; Zbl 0123.02503)] and Peterson (unpublished) the authors prove the following theorem: Let \(\mathfrak X\) be a Schunck class of finite groups and let the finite group \(G = AB = AC = BC\) be the product of two nilpotent subgroups \(A\) and \(B\) and an \(\mathfrak X\)-subgroup \(C\). If for every common prime divisor \(p\) of the orders of \(A\) and \(B\) the group of order \(p\) belongs to \(\mathfrak X\), then \(G\) is a (soluble) \(\mathfrak X\)-group. This theorem becomes false if only one of the subgroups \(A\) and \(B\) is nilpotent (and the other is an \(\mathfrak X\)-group). However the authors prove that if \(\mathfrak X\) is a Schunck class of finite soluble groups (or a saturated formation of finite groups) and the finite group \(G = AB = AK = BK\) is the product of two \(\mathfrak X\)-subgroups \(A\) and \(B\) and a nilpotent normal subgroup \(K\), then \(G\) is an \(\mathfrak X\)-group. An example is exhibited to show that the latter result is no longer true for arbitrary Schunck classes. Finally, in the last section this result is extended to some particular classes of infinite groups.
scientific article; zbMATH DE number 645141
scientific article