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

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.

Abstract licence: CC BY

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