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

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

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.
Year 2020, Volume: 49 Issue: 5, 1706 - 1717, 06.10.2020
https://doi.org/10.15672/hujms.573548

Abstract

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.
There are 42 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Mathematics
Authors

Muhammad Tanveer Hussain This is me 0000-0003-0884-3056

Chenchen Cao 0000-0003-3891-9840

Li Zhang This is me 0000-0002-3132-744X

Project Number 11771409
Publication Date October 6, 2020
Published in Issue Year 2020 Volume: 49 Issue: 5

Cite

APA Hussain, M. T., Cao, C., & Zhang, L. (2020). Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups. Hacettepe Journal of Mathematics and Statistics, 49(5), 1706-1717. https://doi.org/10.15672/hujms.573548
AMA Hussain MT, Cao C, Zhang L. Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups. Hacettepe Journal of Mathematics and Statistics. October 2020;49(5):1706-1717. doi:10.15672/hujms.573548
Chicago Hussain, Muhammad Tanveer, Chenchen Cao, and Li Zhang. “Finite Groups With Given Weakly $\tau_{\sigma}$-Quasinormal Subgroups”. Hacettepe Journal of Mathematics and Statistics 49, no. 5 (October 2020): 1706-17. https://doi.org/10.15672/hujms.573548.
EndNote Hussain MT, Cao C, Zhang L (October 1, 2020) Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups. Hacettepe Journal of Mathematics and Statistics 49 5 1706–1717.
IEEE M. T. Hussain, C. Cao, and L. Zhang, “Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups”, Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 5, pp. 1706–1717, 2020, doi: 10.15672/hujms.573548.
ISNAD Hussain, Muhammad Tanveer et al. “Finite Groups With Given Weakly $\tau_{\sigma}$-Quasinormal Subgroups”. Hacettepe Journal of Mathematics and Statistics 49/5 (October 2020), 1706-1717. https://doi.org/10.15672/hujms.573548.
JAMA Hussain MT, Cao C, Zhang L. Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups. Hacettepe Journal of Mathematics and Statistics. 2020;49:1706–1717.
MLA Hussain, Muhammad Tanveer et al. “Finite Groups With Given Weakly $\tau_{\sigma}$-Quasinormal Subgroups”. Hacettepe Journal of Mathematics and Statistics, vol. 49, no. 5, 2020, pp. 1706-17, doi:10.15672/hujms.573548.
Vancouver Hussain MT, Cao C, Zhang L. Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups. Hacettepe Journal of Mathematics and Statistics. 2020;49(5):1706-17.