On weakly Π-embedded subgroups of finite groups

ZHANG Li; LIU Yufeng

doi:10.3969/j.issn.0253-2778.2016.12.001

JUSTC > 2016 > 46(12): 969-975. > DOI: 10.3969/j.issn.0253-2778.2016.12.001

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On weakly Π-embedded subgroups of finite groups

ZHANG Li^,,
LIU Yufeng

1.
School of Mathematical Sciences, University of Science and Technology of China, Hefei 230026, China
2.
School of Mathematics and Informational Science, Shandong Institute of Business and Technology, Yantai 264005, China

Cite this: JUSTC, 2016, 46(12): 969-975

https://doi.org/10.3969/j.issn.0253-2778.2016.12.001

Funds: Supported by NNSF of China (11371335).

More Information

Corresponding author:
ZHANG Li (corresponding author), female, born in 1991, PhD. Research field: Group theory. E-mail: zhang12@mail.ustc.edu.cn
Received Date: January 07, 2016
Revised Date: May 09, 2016
Accepted Date: May 09, 2016
Published Date: December 29, 2016

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Abstract

Abstract

Let G be a finite group and H a subgroup of G. H is called weakly Π-embedded in G if there exists a subgroup pair (T, S), where T is a quasinormal subgroup of G containing HG and S/HG≤H/HG satisfies Π-property in G/HG, such that |G:HT| is a power of a prime and (H∩T)/HG≤S/HG. Here weakly Π-embedded subgroups were used to explore the structure of finite groups. In particular, new criterions of hypercyclically embedded subgroups were obtained.

Abstract

Let G be a finite group and H a subgroup of G. H is called weakly Π-embedded in G if there exists a subgroup pair (T, S), where T is a quasinormal subgroup of G containing HG and S/HG≤H/HG satisfies Π-property in G/HG, such that |G:HT| is a power of a prime and (H∩T)/HG≤S/HG. Here weakly Π-embedded subgroups were used to explore the structure of finite groups. In particular, new criterions of hypercyclically embedded subgroups were obtained.

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References (22)

References

[1]	DOERK K, HAWKES T. Finite Soluble Groups[M]. Berlin: Walter de Gruyter, 1992.
[2]	GUO W. Structure Theory for Canonical Classes of Finite Groups[M]. Berlin/ Heidelberg: Spring, 2015.
[3]	HUPPERT B. Endliche Gruppen Ⅰ[M]. Berlin/ New York: Springer-Verlag, 1967.
[4]	LI B. On Π-property and Π-normality of subgroups of finite groups[J]. J Algebra, 2011, 334: 321-337.
[5]	CHEN X, GUO W. On Π-supplemented subgroups of a finite group[J]. Comm Algebra, 2016, 44: 731-745.
[6]	LI B. Finite groups with Π-supplemented minimal subgroups[J]. Comm Algebra, 2013, 41: 2 060-2 070.
[7]	SU N, LI Y, WANG Y. A criterion of p-hypercyclically embedded subgroups of finite groups[J]. J Algebra, 2014, 400: 82-93.
[8]	DESKINS W E. On quasinormal subgroups of finite groups[J]. Math Z, 1963, 82: 125-132.
[9]	HUPPERT B, BLACKBURN N. Finite Groups Ⅲ[M]. Berlin/ Heidelberg: Springer-Verlag, 1982.
[10]	MAIER R, SCHMID P. The embedding of quasinormal subgroups in finite groups[J]. Math Z, 1973, 131: 269-272.
[11]	BALLESTER-BOLINCHES A, ESTEBAN-ROMERO R, ASAAD M. Products of Finite Groups[M]. Berlin/ New York: Walter de Gruyter, 2010.
[12]	GUO W, SKIBA A N. On FΦ*-hypercentral subgroups of finite groups[J]. J Algebra, 2012, 372: 275-292.
[13]	LI B, GUO W. On some open problems related to X-permutability of subgroups[J]. Comm Algebra, 2011, 39: 757-771.
[14]	GAGEN T M. Topics in Finite Groups[M]. Melbourne/ New York/ London: Cambridge, 1976.
[15]	GUO W, SKIBA A N. On the intersection of the F-maximal subgroups and the generalized F-hypercentre of a finite group[J]. J Algebra, 2012, 366: 112-125.
[16]	GUO W. The Theory of Classes of Groups[M]. Beijing/ New York/ Dordrecht/ Boston/ London: Science Press/ Kluwer Academic Publishers, 2000.
[17]	WANG Y. c-Normality of groups and its properties[J]. J Algebra, 1996, 180: 954-965.
[18]	CHEN X, MAO Y, GUO W. On finite groups with some primary subgroups satisfying partial S-Π-property[J]. Comm Algebra, 2017, 45: 428-436.
[19]	EZQUERRO L M. A contribution to the theory of finite supersolvable group[J]. Rend Sem Mat Univ Padova, 1993, 89: 161-170.
[20]	LI Y, WANG Y, WEI H. The influence of π-quasinormality of some subgroups of a finite group[J]. Arch Math (Basel), 2003, 81: 245-252.
[21]	SRINIVASAN S. Two sufficient conditions for supersolvability of finite groups[J]. Israel J Math, 1980, 35: 210-214.
[22]	WEI H, WANG Y, LI Y. On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups. II[J]. Comm Algebra, 2003, 31: 4 807-4 816.

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References

[1]	DOERK K, HAWKES T. Finite Soluble Groups[M]. Berlin: Walter de Gruyter, 1992.
[2]	GUO W. Structure Theory for Canonical Classes of Finite Groups[M]. Berlin/ Heidelberg: Spring, 2015.
[3]	HUPPERT B. Endliche Gruppen Ⅰ[M]. Berlin/ New York: Springer-Verlag, 1967.
[4]	LI B. On Π-property and Π-normality of subgroups of finite groups[J]. J Algebra, 2011, 334: 321-337.
[5]	CHEN X, GUO W. On Π-supplemented subgroups of a finite group[J]. Comm Algebra, 2016, 44: 731-745.
[6]	LI B. Finite groups with Π-supplemented minimal subgroups[J]. Comm Algebra, 2013, 41: 2 060-2 070.
[7]	SU N, LI Y, WANG Y. A criterion of p-hypercyclically embedded subgroups of finite groups[J]. J Algebra, 2014, 400: 82-93.
[8]	DESKINS W E. On quasinormal subgroups of finite groups[J]. Math Z, 1963, 82: 125-132.
[9]	HUPPERT B, BLACKBURN N. Finite Groups Ⅲ[M]. Berlin/ Heidelberg: Springer-Verlag, 1982.
[10]	MAIER R, SCHMID P. The embedding of quasinormal subgroups in finite groups[J]. Math Z, 1973, 131: 269-272.
[11]	BALLESTER-BOLINCHES A, ESTEBAN-ROMERO R, ASAAD M. Products of Finite Groups[M]. Berlin/ New York: Walter de Gruyter, 2010.
[12]	GUO W, SKIBA A N. On FΦ*-hypercentral subgroups of finite groups[J]. J Algebra, 2012, 372: 275-292.
[13]	LI B, GUO W. On some open problems related to X-permutability of subgroups[J]. Comm Algebra, 2011, 39: 757-771.
[14]	GAGEN T M. Topics in Finite Groups[M]. Melbourne/ New York/ London: Cambridge, 1976.
[15]	GUO W, SKIBA A N. On the intersection of the F-maximal subgroups and the generalized F-hypercentre of a finite group[J]. J Algebra, 2012, 366: 112-125.
[16]	GUO W. The Theory of Classes of Groups[M]. Beijing/ New York/ Dordrecht/ Boston/ London: Science Press/ Kluwer Academic Publishers, 2000.
[17]	WANG Y. c-Normality of groups and its properties[J]. J Algebra, 1996, 180: 954-965.
[18]	CHEN X, MAO Y, GUO W. On finite groups with some primary subgroups satisfying partial S-Π-property[J]. Comm Algebra, 2017, 45: 428-436.
[19]	EZQUERRO L M. A contribution to the theory of finite supersolvable group[J]. Rend Sem Mat Univ Padova, 1993, 89: 161-170.
[20]	LI Y, WANG Y, WEI H. The influence of π-quasinormality of some subgroups of a finite group[J]. Arch Math (Basel), 2003, 81: 245-252.
[21]	SRINIVASAN S. Two sufficient conditions for supersolvability of finite groups[J]. Israel J Math, 1980, 35: 210-214.
[22]	WEI H, WANG Y, LI Y. On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups. II[J]. Comm Algebra, 2003, 31: 4 807-4 816.

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