Finite groups with given weakly $\tau_{\sigma}$-quasinormal subgroups
Yıl 2020,
Cilt: 49 Sayı: 5, 1706 - 1717, 06.10.2020
Muhammad Tanveer Hussain
Chenchen Cao
Li Zhang
Öz
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 $G$.
Destekleyen Kurum
NNSF of China
Kaynakça
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Heidelberg-New York-Dordrecht-London, 2015.
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subgroups, J. Group Theory 18, 191–200, 2015.
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for primary subgroups, J. Algebra, 432, 205–227, 2015.
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permutably embedded, J. Group Theory, 20, 169–183, 2017.
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Math. 185, 443–453, 2018.
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Sin. 34, 1379–1390, 2018.
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subgroups, Commun. Math. Stat. 5, 83–92, 2017.
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321–337, 2011.
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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.
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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.
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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.
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Publ. Math. Debrecen, 91, 489–502, 2017.
Yıl 2020,
Cilt: 49 Sayı: 5, 1706 - 1717, 06.10.2020
Muhammad Tanveer Hussain
Chenchen Cao
Li Zhang
Kaynakça
- [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.