A property of factorizable groups (Q689786)

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).