Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups

Year 2020, Volume: 49 Issue: 5, 1706 - 1717, 06.10.2020

Muhammad Tanveer Hussain Chenchen Cao , Li Zhang

https://doi.org/10.15672/hujms.573548

Abstract

Let $\sigma=\{{\sigma_i|i\in I}\}$ be a partition of the set of all primes $\mathbb{P}$ and $G$ a finite group. A set $\mathcal{H} $ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every non-identity member of $\mathcal{H}$ is a Hall $\sigma_i$-subgroup of $G$ for some $i\in I$ and $\mathcal{H}$ contains exactly one Hall $\sigma_i$-subgroup of $G$ for every $i$ such that $\sigma_i\cap \pi(G)\neq \emptyset$. Let $\tau_{\mathcal{H}}(A)=\{ \sigma_{i}\in \sigma(G)\backslash \sigma(A) \ |\ \sigma(A) \cap \sigma(H^{G})\neq\emptyset$ for a Hall $\sigma_{i}$-subgroup $H\in \mathcal{H}\}$. A subgroup $A$ of $G$ is said to be $\tau_{\sigma}$-permutable or $\tau_{\sigma}$-quasinormal in $G$ with respect to $\mathcal{H}$ if $AH^{x}=H^{x}A$ for all $x\in G$ and $H\in \mathcal{H}$ such that $\sigma(H)\subseteq \tau_{\mathcal{H}}(A)$, and $\tau_{\sigma}$-permutable or $\tau_{\sigma}$-quasinormal in $G$ if $A$ is $\tau_{\sigma}$-permutable in $G$ with respect to some complete Hall $\sigma$-set of $G$. We say that a subgroup $A$ of $G$ is weakly $\tau_{\sigma}$-quasinormal in $G$ if $G$ has a $\sigma$-subnormal subgroup $T$ such that $AT=G$ and $A\cap T\leq A_{\tau_{\sigma}G}$, where $A_{\tau_{\sigma}G}$ is the subgroup generated by all those subgroups of $A$ which are $\tau_{\sigma}$-quasinormal in $G$. We study the structure of $G$ being based on the assumption that some subgroups of $G$ are weakly $\tau_{\sigma}$-quasinormal in $G$.

Keywords

finite groups, $\sigma$-permutable subgroup, $\tau_{\sigma}$-quasinormal subgroup, weakly $\tau_{\sigma}$-quasinormal subgroup, supersoluble group

Supporting Institution

NNSF of China

Project Number

11771409

References

• [1] M. Asaad, On the solvability of finite groups, Arch. Math. 51, 289–293, 1988.
• [2] M. Asaad, On maximal subgroups of Sylow subgroups of finite groups, Comm. Algebra 26, 3647–3652, 1998.
• [3] M. Asaad, M. Ramadan and A. Shaalan, Influence of $\pi$-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group, Arch. Math. 56, 521–527, 1991.
• [4] A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of Finite Groups, Walter de Gruyter, Berlin, 2010.
• [5] A. Ballester-Bolinches and M.C. Pedraza-Aguilera, On minimal subgroups of finite groups, Acta Math. Hungar. 73, 335–342, 1996.
• [6] J.C. Beidleman and A.N. Skiba, On $\tau_{\sigma}$-quasinormal subgroups of finite groups, J. Group Theory, 20, 955–969, 2017.
• [7] J. Buckley, Finite groups whose minimal subgroups are normal, Math. Z. 116, 15–17, 1970.
• [8] C. Cao, Z. Wu and W. Guo, Finite groups with weakly $\sigma$-permutable subgroups, Siberian Math. J. 59, 157–165, 2018.
• [9] X. Chen, W. Guo and A.N. Skiba, Some conditions under which a finite group belongs to a Baer-local formation, Comm. Algebra, 42, 4188–4203, 2014.
• [10] W.E. Deskins, On quasinormal subgroups of finite groups, Math. Z. 82, 125–132, 1963.
• [11] K. Doerk and T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin-New York, 1992.
• [12] D. Gorenstein, Finite Groups, Harper and Row Publishers, New York-Evanston- London, 1968.
• [13] W. Guo, The Theory of Classes of Groups, Science Press-Kluwer Academic Publish- ers, Dordrecht-Boston-London, 2000.
• [14] W. Guo, Structure Theory for Canonical Classes of Finite Groups, Springer, Heidelberg-New York-Dordrecht-London, 2015.
• [15] W. Guo and A.N. Skiba, Finite groups with permutable complete Wielandt sets of subgroups, J. Group Theory 18, 191–200, 2015.
• [16] W. Guo and A.N. Skiba, Finite groups with generalized Ore supplement conditions for primary subgroups, J. Algebra, 432, 205–227, 2015.
• [17] W. Guo and A.N. Skiba, Groups with maximal subgroups of Sylow subgroups $\sigma$- permutably embedded, J. Group Theory, 20, 169–183, 2017.
• [18] W. Guo and A.N. Skiba, On $\Pi$-quasinormal subgroups of finite groups, Monatsh. Math. 185, 443–453, 2018.
• [19] W. Guo and A.N. Skiba, On $\sigma$-semipermutable subgroups of finite groups, Acta Math. Sin. 34, 1379–1390, 2018.
• [20] W. Guo, C. Cao, A.N. Skiba and D.A. Sinitsa, Finite groups with ${\mathcal{H}}$-permutable subgroups, Commun. Math. Stat. 5, 83–92, 2017.
• [21] B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin-Heidelberg-New York, 1967.
• [22] B. Huppert and N. Blackburn, Finite groups III, Springer-verlag, Berlin-Heidelberg- New York, 1982.
• [23] B. Li, On $\Pi$-property and $\Pi$-normality of subgroups of finite groups, J. Algebra, 334, 321–337, 2011.
• [24] D. Li and X. Guo, The influence of c-normality of subgroups on the structure of finite groups II, Commun. Algebra, 26, 1913–1922, 1998.
• [25] Y. Li, Y. Wang and H. Wei, The influence of $\pi$-quasinormality of some subgroups of a finite groups, Arch. Math. 81, 245–252, 2003.
• [26] V.O. Lukyanenko and A.N. Skiba, On weakly $\tau$-quasinormal subgroups of finite groups, Acta Math. Hungar. 125, 237–248, 2009.
• [27] V.O. Lukyanenko and A.N. Skiba, Finite groups in which $\tau$-quasinormality is a tran- sitive relation, Rend. Sem. Mat. Univ. Padova, 124, 231–246, 2010.
• [28] L. Miao, On weakly s-permutable subgroups of finite groups, Bull. Barz. Math. Soc. 41 (2), 223–235, 2010.
• [29] M. Ramadan, Influence of normality on maximal subgroups of Sylow subgroups of a finite group, Acta Math. Hungar. 59, 107–110, 1992.
• [30] R. Schmidt, Subgroup Lattices of Groups, Walter de Gruyter, Berlin, 1994.
• [31] A.N. Skiba, On weakly s-permutable subgroups of finite groups, J. Algebra, 315, 192– 209, 2007.
• [32] A.N. Skiba, On two questions of L. A. Shemetkov concerning hypercyclically embedded subgroups of finite groups, J. Group Theory, 13, 841–850, 2010.
• [33] A.N. Skiba, A characterization of the hypercyclically embedded subgroups of finite group, J. Pure Appl. Algebra, 215, 257–261, 2011.
• [34] A.N. Skiba, On $\sigma$-subnormal and $\sigma$-permutable subgroups of finite groups, J. Algebra, 436, 1–16, 2015.
• [35] A.N. Skiba, On some results in the theory of finite partially soluble groups, Commun. Math. Stat. 4, 281–309, 2016.
• [36] A.N. Skiba, Some characterizations of finite $\sigma$-soluble $P\sigmaT$-groups, J. Algebra, 495, 114–129, 2018.
• [37] S. Srinivasan, Two sufficient conditions for supersolvability of finite groups, Israel J. Math. 35, 210–214, 1980.
• [38] Y. Wang, c-normality of groups and its properties, J. Algebra, 180, 954–965, 1996.
• [39] H. Wei, On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups, Comm. Algebra, 29, 2193–2200, 2001.
• [40] H. Wei, Y. Wang and Y. Li, On c-normal maximal and minimal subgroups of Sylow subgroups of finite groups II, Comm. Algebra, 31, 4807–4816, 2003.
• [41] M. Weinstein et al., Between Nilpotent and Solvable, Polygonal Publishing House, Washington, 1982.
• [42] C. Zhang, Z. Wu and W. Guo, On weakly $\sigma$-permutable subgroups of finite groups, Publ. Math. Debrecen, 91, 489–502, 2017.