Abstract
Abstract In this paper, we call a finite group G G an N L M NLM -group ( N C M NCM -group, respectively) if every non-normal cyclic subgroup of prime order or order 4 (prime power order, respectively) in G G is contained in a non-normal maximal subgroup of G G . Using the property of N L M NLM -groups and N C M NCM -groups, we give a new necessary and sufficient condition for G G to be a solvable T T -group (normality is a transitive relation), some sufficient conditions for G G to be supersolvable, and the classification of those groups whose all proper subgroups are N L M NLM -groups.
Highlights
All groups considered in this paper are finite and our notation is standard
Kaplan [8] introduced the following definition in investigating solvable T -groups non-normal subgroups do not seem to be directly related to solvable T -groups
We say that G has the NNM property – shortly: G is an NNM-group – if each non-normal subgroup of G is contained in a non-normal maximal subgroup of G
Summary
All groups considered in this paper are finite and our notation is standard. Normal subgroups play a crucial role in investigating group structures. There are many good results in characterizing the structure of finite groups by applying some given properties of non-normal subgroups, for instance, [1,2,3,4,5,6,7,8]. We say that G has the NNM property – shortly: G is an NNM-group – if each non-normal subgroup of G is contained in a non-normal maximal subgroup of G. It is natural to consider the structure of a group G in which only a part of non-normal subgroups is contained in a non-normal maximal subgroup of G. We say that G has the NCM property – shortly: G is an NCM-group – if each non-normal cyclic subgroup of prime power order in G is contained in a non-normal maximal subgroup of G
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