Groups decomposable into quasicentralizer products
The quasicentralizer of a subgroup A in a group G is, by definition, the intersection Q(A) of the normalizers in G of all subgroups of A. A group G is said to be a quasicentralizer product of a subgroup A on a subgroup B if \(G=AB\) and \(B\subset Q(A)\). Periodic groups decomposable into a quasicentralizer product with locally nilpotent or abelian factors are studied in this article. It is proved that if G is a quasicentralizer product of A on B where A and B are periodic locally nilpotent subgroups then either G is locally nilpotent or the locally nilpotent coradical of G is a non-trivial abelian involution-free subgroup which is complementable in G and decomposable into the direct product of all Sylow subgroups of A which have trivial intersections with the center of G (Theorem 1). A periodic group G is decomposable into a quasi-centralizer product with abelian factors if and only if G is a semi-direct product of an abelian subgroup N all of whose subgroups are normal in G on a hypercentral subgroup M which is a quasicentralizer product of an abelian subgroup C on an abelian subgroup Q such that C is normal in G and elements in C and N have coprime orders (Theorem 3).
scientific article; zbMATH DE number 3948598