IJPAM: Volume 13, No. 3 (2004)

M-GROUP AND SEMI-DIRECT PRODUCT

Liguo He
Department of Mathematics
Shenyang University of Technology
Shenyang, 110023, P.R. CHINA
e-mail: heliguo@online.ln.cn


Abstract.A finite group is said to be an Mim-group if all of its subgroups are M-groups. We show that a group is an M-group if it is the semidirect product of an Abelian group with an Mim-group. A group is an M-group if it is the semidirect product of a Sylow tower group with an Mim-group such that their orders are coprime and the Sylow subgroups of the Sylow tower group are Abelian. We also show that if $G$ is an inner supersolvable group, then $G/ \Phi$(G) ($\Phi(G)$, Frattini subgroup) is an M-group. Generally, the inner supersolvable groups need not be M-groups.

Received: March 27, 2004

AMS Subject Classification: 20C15

Key Words and Phrases: M-group, Mim-group, semi-direct product of groups, supersolvable group

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2004
Volume: 13
Issue: 3