Structure of finite centralizer-factorizable groups (Q762268)

A group G is said to be centralizer-factorable if the centralizer of each subgroup of G has a complement in G. The present paper provides a complete description of finite centralizer-factorable groups. Such groups are semidirect products \(\prod_{i}A_ i\leftthreetimes \prod_{j}<b_ j>\) of direct products \(\prod_{i}A_ i\) and \(\prod_{j}<b_ j>\) where each subgroup \(A_ k\leftthreetimes <b_{\ell}>\) is either abelian or a Frobenius group (we have omitted some additional conditions). The proof is heavily based on the previous work of the author [Preprint Akad. Nauk USSR, Inst. Math. No.82.47 (1982)].