# On $\Phi$-$\tau$-quasinormal subgroups of finite groups

Document Type: Research Paper

Authors

1 Institute of Quantum Information Science‎, ‎Shanxi Datong University‎ ‎Datong 037009‎, ‎P.R‎. ‎China.

2 School of Mathematics and Computer‎, ‎University of Datong of Shanxi‎, ‎Datong 037009‎, ‎P.R‎. ‎China.

3 School of Mathematical Sciences‎, ‎University of Science and Technology of China‎, ‎Hefei‎, ‎230026‎, ‎P.R‎. ‎China.

4 School of mathematics and statistics‎, ‎Jiangsu Normal University‎ ‎Xuzhou‎, ‎221116‎, ‎P.R‎. ‎China.

Abstract

‎Let $\tau$ be a subgroup functor and $H$ a $p$-subgroup of a finite group $G$‎. ‎Let $\bar{G}=G/H_{G}$ and $\bar{H}=H/H_{G}$‎. ‎We say that $H$ is $\Phi$-$\tau$-quasinormal in $G$ if for some $S$-quasinormal subgroup $\bar{T}$ of $\bar{G}$ and some $\tau$-subgroup $\bar{S}$ of $\bar{G}$ contained in $\bar{H}$‎, ‎$\bar{H}\bar{T}$ is $S$-quasinormal in $\bar{G}$ and $\bar{H}\cap\bar{T}\leq \bar{S}\Phi(\bar{H})$‎. ‎In this paper‎, ‎we study the structure of a group $G$ under the condition that some primary subgroups of $G$ are $\Phi$-$\tau$-quasinormal in $G$‎. ‎Some new characterizations about $p$-nilpotency and solubility of finite groups are obtained.

Keywords

Main Subjects

### References

M. Asaad, Finite groups with given nearly s-embedded subgroups, Acta Math. Hungar. 144 (2014), no. 2, 499--514.

A. Ballester-Bolinches and R. Esteban-Romero, On minimal non-supersoluble groups, Rev. Mat. Iberoam. 23 (2007), no. 1, 127--142.

Z. Chen, On a theorem of Srinivasan, J. Southwest Normal Univ. (Nat. Sci.) 12 (1987), no. 1, 1--4.

X. Chen, W. Guo and A. N. Skiba, Some conditions under which a finite group belongs to a Baer-local formation, Comm. Algebra 42 (2014) 4188--4203.

X. Chen, W. Guo and A. N. Skiba, F-Embedded and F-Embedded Subgroups of Finite Groups, Algebra Logic 54 (2015), no. 3, 226--244.

K. Doerk and T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin-New York, 1992.

D. Gorenstein, Finite Groups, Harper & Row, 1968.

W. Guo, The Theory of Classes of Groups, Springer Netherlands, 2000.

W. Guo, Structure Theory for Cononical Classes of Finite Groups, Springer-Verlag, Berlin-Heidelburg, 2015.

W. Guo and A. N. Skiba, Finite groups with given s-embedded and n-embedded subgroups, J. Algebra 321 (2009) 2843--2860.

W. Guo and A. N. Skiba, Finite groups with generalized Ore supplement conditions for primary subgroups, J. Algebra 432 (2015) 205--227.

L. Huo and W. Guo, On nearly SS-embedded subgroups of finite groups, Chin. Ann. Math. Ser. B 35 (2014), no. 6, 885--894.

B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin-Heidelberg, 1967.

B. Huppert, N. Blackburn, Finite Groups III, Spring-Verlag, Berlin-Heidelberg, 1982.

X. Li and T. Zhao, S-supplemented subgroups of finite groups, Ukrainian Math. J. 64 (2012), no. 1, 102--109.

I.A. Malinowska, Finite groups with sn-embedded or s-embedded subgroups, Acta Math. Hungar. 136 (2012), no. 1-2, 76--89.

A.N. Skiba, On two questions of L.A. Shemetkov concerning hypercyclically embedded subgroups of finite groups, J. Group Theory 13 (2010) 841--850.

Y. Wang, Finite groups with some subgroups of Sylow subgroups c-supplemented, J. Algebra 224 (2000) 467--478.

Y. Wang and W. Guo, Nearly s-normality of groups and its properties, Comm. Algebra 38 (2010), no. 10, 3821--3836.

### History

• Receive Date: 15 January 2016
• Revise Date: 17 December 2016
• Accept Date: 15 January 2017