MATHEMATICS

On a finite group factorized by a $B$-group and a $z$-group

V. N. Kniahina

Francisk Skorina Gomel State University

Abstract: A finite non-nilpotent group is called a $B$-group if all proper subgroups of its quotient group by the Frattini subgroup are primary. A finite group whose Sylow subgroups are all cyclic is called a $z$-group. We study a finite group $G$ that can be represented as a product of its $B$-subgroup and $z$-subgroup of coprime orders. For a solvable groups $G$, we prove that the second derived subgroup is nilpotent, the derivative length of the quotient group by the Frattini subgroup does not exceed three, and the $p$-length is at most two. If $G$ is a simple group, then $G$ is isomorphic to determined.

Keywords: finite group, $B$-group, $z$-group, $p$-length, derivative length, factorizable group.

UDC: 512.542

Received: 15.08.2025

DOI: 10.54341/20778708_2025_4_65_67