Skip to main content Accessibility help
×
Home

On the ℱϕ-Hypercentre of Finite Groups

  • Juping Tang (a1) and Long Miao (a1)

Abstract

Let $G$ be a finite group and let $\mathcal{F}$ be a class of groups. Then ${{Z}_{\mathcal{F}\Phi }}\left( G \right)$ is the $\mathcal{F}\Phi$ -hypercentre of $G$ , which is the product of all normal subgroups of $G$ whose non-Frattini $G$ -chief factors are $\mathcal{F}$ -central in $G$ . A subgroup $H$ is called $\mathcal{M}$ -supplemented in a finite group $G$ if there exists a subgroup $B$ of $G$ such that $G\,=\,HB\,\text{and}\,{{H}_{1}}B$ is a proper subgroup of $G$ for any maximal subgroup ${{H}_{1}}$ of $H$ . The main purpose of this paper is to prove the following: Let $E$ be a normal subgroup of a group $G$ . Suppose that every noncyclic Sylow subgroup $P\,\text{of}\,{{F}^{*}}\left( E \right)$ has a subgroup $D$ such that $1\,<\,\left| D \right|\,<\left| P \right|$ and every subgroup $H\,\text{of}\,P$ with order $\left| H \right|\,=\,\left| D \right|$ is $\mathcal{M}$ -supplemented in $G$ , then $E\,\le \,{{Z}_{\mathcal{U}\Phi }}\left( G \right)$ .

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

      Find out more about the Kindle Personal Document Service.

      On the ℱϕ-Hypercentre of Finite Groups
      Available formats
      ×

      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

      On the ℱϕ-Hypercentre of Finite Groups
      Available formats
      ×

      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

      On the ℱϕ-Hypercentre of Finite Groups
      Available formats
      ×

Copyright

Corresponding author

L. Miao is the corresponding author

Footnotes

Hide All

This research is supported by NSFC (Grant #11271016), the Postgraduate Innovation Project of Jiangsu Province (No. CXZZ13–0890), and the Natural Science Fund for Colleges and Universities in Anhui Province (Grant #KJ2013B138).

Footnotes

References

Hide All
[1] Asaad, M., Finite groups with certain subgroups of Sylow subgroups complemented. J. Algebra 323 (2010), no. 7, 19581965. http://dx.doi.org/10.1016/j.jalgebra.2010.02.006
[2] Ballester-Bolinches, A., Wang, Y., and Guo, X., C-supplemented subgroups of finite groups. Glasg. Math. J. 42 (2000), no. 3, 383389. http://dx.doi.org/10.1017/S001708950003007X
[3] Doerk, K. and Hawkes, T., Finite soluble groups. de Gruyter Expositions in Mathematics, 4, Walter de Gruyter, Berlin, 1992.
[4] Guo, W., The theory of classes of groups. Mathematics and its Applications, 505, Kluwer Academic Publishers Group, Dordrecht; Science Press, Beijing, 2000.
[5] Hall, P., A characteristic property of soluble groups. J. London Math. Soc. 12 (1937), 189200. http://dx.doi.org/10.1112/jlms/s1-12.2.198
[6] Huppert, B., Endliche Gruppen. I Die Grundlehren der MathematischenWissenschaften, 134, Springer-Verlag, Berlin-New York, 1967.
[7] Huppert, B. and Blackburn, N., Finite groups. III. Grundlehren der MathematischenWissenschaften, 243, Springer-Verlag, Berlin-New York, 1982.
[8] Li, S. and He, X., On normally embedded subgroups of prime power order in finite groups. Comm. Algebra 36 (2008), no. 6, 23332340. http://dx.doi.org/10.1080/00927870701509370
[9] Miao, L. and Lempken, W., OnM-supplemented subgroups of finite groups. J. Group Theory 12 (2009), no. 2, 271289.
[10] Monakhov, V. S. and Shnyparkov, A. V., On the p-supersolubility of a finite group with a-complemented Sylow p-subgroup. Sib. Math. J. 50 (2009), no. 4, 681686.
[11] Shemetkov, L. A. and Skiba, A. N., On the X-hypercentre of finite groups. J. Algebra 322 (2009), no. 6, 21062117. http://dx.doi.org/10.1016/j.jalgebra.2009.03.029
[12] Shemetkov, L. A., Formations of finite groups. (Russian) Nauka, Moscow, 1978.
[13] Skiba, A. N., On weakly s-permutable subgroups of finite groups. J. Algebra 315 (2007), no. 1, 192209. http://dx.doi.org/10.1016/j.jalgebra.2007.04.025
[14] Srinivasan, S., Two sufficient conditions for supersolvability of finite groups. Israel J. Math. 35 (1980), no. 3, 210214. http://dx.doi.org/10.1007/BF02761191
[15] Wang, Y., Finite groups with some subgroups of Sylow subgroups c-supplemented. J. Algebra 224 (2000), no. 2, 467478. http://dx.doi.org/10.1006/jabr.1999.8079
[16] Wang, Y., Wei, H., and Li, Y., A generalization of Kramer's theorem and its applications. Bull. Aust. Math. Soc. 65 (2002), no. 3, 467475. http://dx.doi.org/10.1017/S0004972700020517
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

On the ℱϕ-Hypercentre of Finite Groups

  • Juping Tang (a1) and Long Miao (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed