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 g...

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Main Authors:	Guo Pengfei, Han Zhangjia
Format:	article
Language:	EN
Published:	De Gruyter 2021
Subjects:	normal subgroups minimal subgroups maximal subgroups supersolvable groups t-groups 20e34 20f16 Mathematics QA1-939
Online Access:	https://doaj.org/article/93bfaa4ff88f45be96cf359eed29f607
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spelling	oai:doaj.org-article:93bfaa4ff88f45be96cf359eed29f6072021-12-05T14:10:52ZOn non-normal cyclic subgroups of prime order or order 4 of finite groups2391-545510.1515/math-2021-0012https://doaj.org/article/93bfaa4ff88f45be96cf359eed29f6072021-03-01T00:00:00Zhttps://doi.org/10.1515/math-2021-0012https://doaj.org/toc/2391-5455In 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.Guo PengfeiHan ZhangjiaDe Gruyterarticlenormal subgroupsminimal subgroupsmaximal subgroupssupersolvable groupst-groups20e3420f16MathematicsQA1-939ENOpen Mathematics, Vol 19, Iss 1, Pp 63-68 (2021)
institution	DOAJ
collection	DOAJ
language	EN
topic	normal subgroups minimal subgroups maximal subgroups supersolvable groups t-groups 20e34 20f16 Mathematics QA1-939
spellingShingle	normal subgroups minimal subgroups maximal subgroups supersolvable groups t-groups 20e34 20f16 Mathematics QA1-939 Guo Pengfei Han Zhangjia On non-normal cyclic subgroups of prime order or order 4 of finite groups
description	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.
format	article
author	Guo Pengfei Han Zhangjia
author_facet	Guo Pengfei Han Zhangjia
author_sort	Guo Pengfei
title	On non-normal cyclic subgroups of prime order or order 4 of finite groups
title_short	On non-normal cyclic subgroups of prime order or order 4 of finite groups
title_full	On non-normal cyclic subgroups of prime order or order 4 of finite groups
title_fullStr	On non-normal cyclic subgroups of prime order or order 4 of finite groups
title_full_unstemmed	On non-normal cyclic subgroups of prime order or order 4 of finite groups
title_sort	on non-normal cyclic subgroups of prime order or order 4 of finite groups
publisher	De Gruyter
publishDate	2021
url	https://doaj.org/article/93bfaa4ff88f45be96cf359eed29f607
work_keys_str_mv	AT guopengfei onnonnormalcyclicsubgroupsofprimeorderororder4offinitegroups AT hanzhangjia onnonnormalcyclicsubgroupsofprimeorderororder4offinitegroups
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On non-normal cyclic subgroups of prime order or order 4 of finite groups

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