On weakly 𝓗-permutable subgroups of finite groups
-
Chenchen Cao
,
Venus Amjid
Abstract
Let σ = {σi ∣i ∈ I} be some partition of the set of all primes ℙ, G be a finite group and σ(G) = {σi∣σi ∩ π(G) ≠ ∅}. G is said to be σ-primary if ∣σ(G)∣ ≤ 1. A subgroup H of G is said to be σ-subnormal in G if there exists a subgroup chain H = H0 ≤ H1 ≤ … ≤ Ht = G such that either Hi−1 is normal in Hi or Hi/(Hi−1)Hi is σ-primary for all i = 1, …, t. A set 𝓗 of subgroups of G is said to be a complete Hallσ-set of G if every non-identity member of 𝓗 is a Hall σi-subgroup of G for some i and 𝓗 contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). Let 𝓗 be a complete Hall σ-set of G. A subgroup H of G is said to be 𝓗-permutable if HA = AH for all A ∈ 𝓗. We say that a subgroup H of G is weakly 𝓗-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ H𝓗, where H𝓗 is the subgroup of H generated by all those subgroups of H which are 𝓗-permutable.
By using the weakly 𝓗-permutable subgroups, we establish some new criteria for a group G to be σ-soluble and supersoluble, and we also give the conditions under which a normal subgroup of G is hypercyclically embedded.
This work was supported by the NNSF of China (11771409), Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences and Anhui Initiative in Quantum Information Technologies (AHY150200)
(Communicated by Vincenzo Marra )
References
[1] Assad, M.: On maximal subgroups of Sylow subgroups of finite groups, Comm. Algebra 26(11) (1998), 3647–3652.10.1080/00927879808826364Suche in Google Scholar
[2] Assad, M.: Finite groups with certain subgroups of Sylow subgroups complemented, J. Algebra 323 (2010), 1958–1965.10.1016/j.jalgebra.2010.02.006Suche in Google Scholar
[3] Assad, M.—Ramadan, M.—Shaalan, A.: Influence ofπ-quasinormality on maximal subgroups of Sylow subgroups of Fitting subgroup of a finite group, Arch. Math. (Basel) 56 (1991), 521–527.10.1007/BF01246766Suche in Google Scholar
[4] Ballester-Bolinches, A.—Esteban-Romero, R.—Asaad, M.: Products of Finite Groups, Walter de Gruyter, Berlin, New York, 2010.10.1515/9783110220612Suche in Google Scholar
[5] Ballester-Bolinches, A.—Wang, Y.—Guo, X.: c-supplemented subgroups of finite groups, Glasgow Math. J. 42 (2000), 383–389.10.1017/S001708950003007XSuche in Google Scholar
[6] Chen, X.—Guo, W.—Skiba, A. N.: Some conditions under which a finite group belongs to a Baer-local formation, Comm. Algebra 42 (2014), 4188–4203.10.1080/00927872.2013.806519Suche in Google Scholar
[7] Doerk, K.—Hawkes, T.: Finite Soluble Groups, Walter de Gruyter, Berlin, 1992.10.1515/9783110870138Suche in Google Scholar
[8] Guo, W.: The Theory of Classes of Groups, Science Press-Kluwer Academic Publishers, Beijing, New York, Dordrecht, Boston, London, 2000.Suche in Google Scholar
[9] Guo, W.: Structure Theory for Canonical Classes of Finite Groups, Springer, New York, London, 2015.10.1007/978-3-662-45747-4Suche in Google Scholar
[10] Guo, W.—Cao, C.—Skiba, A. N.—Sinitsa, D. A.: Finite groups with 𝓗-permutable subgroups, Commun. Math. Stat. 5 (2017), 83–92.10.1007/s40304-017-0101-1Suche in Google Scholar
[11] Guo, W.—Skiba, A. N.: Finite groups with generalized Ore supplement conditions for primary subgroups, J. Algebra 432 (2015), 205–227.10.1016/j.jalgebra.2015.02.025Suche in Google Scholar
[12] Guo, W.—Skiba, A. N.: Finite groups with permutable complete Wielandt set of subgroups, J. Group Theory 18 (2015), 191–200.10.1515/jgth-2014-0045Suche in Google Scholar
[13] Guo, W.—Skiba, A. N.: On Π-permutable subgroups of finite groups, Monatsh. Math. 185 (2018), 443–453.10.1007/s00605-016-1007-9Suche in Google Scholar
[14] Huppert, B.: Endliche GruppenI, Springer-Verlag, Berlin, 1967.10.1007/978-3-642-64981-3Suche in Google Scholar
[15] Knyagina, V. N.—Monakhov, V. S.: On theπ′-properties of a finite group possessing a Hallπ-subgroup, Sib. Math. J. 52(2) (2011), 297–309.10.1134/S0037446611020066Suche in Google Scholar
[16] Li, B.: On Π-property and Π-normality of subgroups of finite groups, J. Algebra 334 (2011), 321–337.10.1016/j.jalgebra.2010.12.018Suche in Google Scholar
[17] Miao, L.: On weakly s-permutable subgroups of finite groups, Bull. Braz. Math. Soc. 41(2) (2010), 223–235.10.1007/s00574-010-0011-2Suche in Google Scholar
[18] Schmidt, R.: Subgroups Lattices of Groups, Walter de Gruyter, Berlin, 1994.10.1515/9783110868647Suche in Google Scholar
[19] Skiba, A. N.: Onσ-subnormal andσ-permutable subgroups of finite groups, J. Algebra 436 (2015), 1–16.10.1016/j.jalgebra.2015.04.010Suche in Google Scholar
[20] Skiba, A. N.: A generalization of Hall theorem, J. Algebra Appl. 15(4) (2015), 21–36.10.1142/S0219498816500857Suche in Google Scholar
[21] Skiba, A. N.: A characterization of the hypercyclically embedded subgroups of finite groups, J. Pure Appl. Algebra 215 (2011), 257–261.10.1016/j.jpaa.2010.04.017Suche in Google Scholar
[22] Skiba, A. N.: On two questions of L.A. Shemetkov concerning hypercyclically embeded subgroups of finite groups, J. Group Theory 13 (2010), 841–850.10.1515/jgt.2010.027Suche in Google Scholar
[23] Skiba, A. N.: On weaklys-permutable subgroups of finite groups, J. Algebra 315 (2007), 192–209.10.1016/j.jalgebra.2007.04.025Suche in Google Scholar
[24] Skiba, A. N.: On some results in the theory of finite partially soluble groups, Commun. Math. Stat. 4 (2016), 281–309.10.1007/s40304-016-0088-zSuche in Google Scholar
[25] Srinivasan, S.: Two sufficient conditions for supersolvability of finite groups, Isr. J. Math. 35(3) (1980), 210–214.10.1007/BF02761191Suche in Google Scholar
[26] Wang, Y.: C-normality of groups and its properties, J. Algebra 180 (1996), 954–965.10.1006/jabr.1996.0103Suche in Google Scholar
[27] Wei, H.: Onc-normal maximal and minimal subgroups of Sylow subgroups of finite groups, Comm. Algebra 29(5) (2001), 2193–2200.10.1081/AGB-120023133Suche in Google Scholar
[28] Weinsten, M. et al: Between Nilpotent and Soluble, Polygonal Publishing House, Passaic, 1982.Suche in Google Scholar
[29] Zhang, C.—Wu, Z.—Guo, W.: On weaklyσ-permutable subgroups of finite groups, Publ. Math. Debrecen 91(3–4) (2017), 489–502.10.5486/PMD.2017.7815Suche in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
Artikel in diesem Heft
- Regular papers
- On the finite embeddability property for quantum B-algebras
- A dual Ramsey theorem for finite ordered oriented graphs
- On EMV-Semirings
- A nonsymmetrical matrix and its factorizations
- On weakly 𝓗-permutable subgroups of finite groups
- The left Riemann-Liouville fractional Hermite-Hadamard type inequalities for convex functions
- A geometrical version of Hardy-Rellich type inequalities
- On a Choquet-Stieltjes type integral on intervals
- Uniqueness of meromorphic functions sharing four small functions on annuli
- A subclass of uniformly convex functions and a corresponding subclass of starlike function with fixed coefficient associated with q-analogue of Ruscheweyh operator
- Functions of bounded variation related to domains bounded by conic sections
- Unified solution of Fekete-Szegö problem for subclasses of starlike mappings in several complex variables
- Oscillatory criteria for the second order linear ordinary differential equations
- When deviation happens between rough statistical convergence and rough weighted statistical convergence
- The retraction of certain banach right modules associated to a character
- On sequence spaces generated by binomial difference operator of fractional order
- Some refinements of young type inequality for positive linear map
- On the Gromov-Hausdorff limit of metric spaces
- The beta exponentiated Nadarajah-Haghighi distribution: theory, regression model and application
- Hilbert algebras with supremum generated by finite chains
Artikel in diesem Heft
- Regular papers
- On the finite embeddability property for quantum B-algebras
- A dual Ramsey theorem for finite ordered oriented graphs
- On EMV-Semirings
- A nonsymmetrical matrix and its factorizations
- On weakly 𝓗-permutable subgroups of finite groups
- The left Riemann-Liouville fractional Hermite-Hadamard type inequalities for convex functions
- A geometrical version of Hardy-Rellich type inequalities
- On a Choquet-Stieltjes type integral on intervals
- Uniqueness of meromorphic functions sharing four small functions on annuli
- A subclass of uniformly convex functions and a corresponding subclass of starlike function with fixed coefficient associated with q-analogue of Ruscheweyh operator
- Functions of bounded variation related to domains bounded by conic sections
- Unified solution of Fekete-Szegö problem for subclasses of starlike mappings in several complex variables
- Oscillatory criteria for the second order linear ordinary differential equations
- When deviation happens between rough statistical convergence and rough weighted statistical convergence
- The retraction of certain banach right modules associated to a character
- On sequence spaces generated by binomial difference operator of fractional order
- Some refinements of young type inequality for positive linear map
- On the Gromov-Hausdorff limit of metric spaces
- The beta exponentiated Nadarajah-Haghighi distribution: theory, regression model and application
- Hilbert algebras with supremum generated by finite chains
