On non-normal cyclic subgroups of prime order or order 4 of finite groups

Abstract

In this paper, we call a finite group GG an NLMNLM-group (NCMNCM-group, respectively) if every non-normal cyclic subgroup of prime order or order 4 (prime power order, respectively) in GG is contained in a non-normal maximal subgroup of GG. Using the property of NLMNLM-groups and NCMNCM-groups, we give a new necessary and sufficient condition for GG to be a solvable TT-group (normality is a transitive relation), some sufficient conditions for GG to be supersolvable, and the classification of those groups whose all proper subgroups are NLMNLM-groups.

Keywords

• normal subgroups
• minimal subgroups
• maximal subgroups
• supersolvable groups
• t-groups
• 20e34
• 20f16