A property of factorizable groups. (English) Zbl 0783.20016

If the finite soluble group \(G=AB\) is the product of subgroups \(A\) and \(B\), then it is shown that \(O_ \pi(A)\cap O_ \pi(B)\subseteq O_ \pi(G)\) for every set of primes (Theorem 1). This result is then generalized to more general situations. For instance, a corresponding statement holds when \(G\) is a periodic hyperabelian group with no perfect subgroups and no infinite elementary abelian groups involved in \(O_ \pi(A)\) (Corollary 1).
Reviewer: B.Amberg (Mainz)


20E22 Extensions, wreath products, and other compositions of groups
20F50 Periodic groups; locally finite groups
20E07 Subgroup theorems; subgroup growth
20D40 Products of subgroups of abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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