On Sylow permutable subgroups of finite groups
-
Adolfo Ballester-Bolinches
,
Hermann Heineken
Abstract
A subgroup H of a group G is called Sylow permutable, or S-permutable, in G if H permutes with all Sylow p-subgroups of G for all primes p.
A group G is said to be a PST-group if Sylow permutability is a transitive relation in G.
We show that a group G which is factorised by a normal subgroup and a subnormal PST-subgroup of odd order is supersoluble.
As a consequence, the normal closure
Dedicated to Professor James C. Beidleman on his 80th birthday
Funding source: Ministerio de Economía y Competitividad
Award Identifier / Grant number: MTM2014-54707-C3-1-P
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11271085
Funding source: Natural Science Foundation of Guangdong Province
Award Identifier / Grant number: 2015A030313791
Funding statement: The first and third author have been supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economía y Competitividad, Spain, and FEDER, European Union. The first author has also been supported by a project from the National Natural Science Foundation of China (No. 11271085) and a project of Natural Science Foundation of Guangdong Province (No. 2015A030313791).
References
[1] A. Ballester-Bolinches and J. C. Beidleman, Some special classes of finite soluble PST-groups, Ric. Mat. 64 (2015), no. 2, 325–330. 10.1007/s11587-015-0237-9Search in Google Scholar
[2] A. Ballester-Bolinches, R. Esteban-Romero and M. Asaad, Products of Finite Groups, De Gruyter Exp. Math. 53, Walter de Gruyter, Berlin, 2010. 10.1515/9783110220612Search in Google Scholar
[3] J. C. Beidleman and H. Heineken, On the Fitting core of a formation, Bull. Aust. Math. Soc. 68 (2003), no. 1, 107–112. 10.1017/S0004972700037461Search in Google Scholar
[4] Z. M. Chen, On a theorem of Srinivasan, Southwest Normal Univ. Nat. Sci. 12 (1987), no. 1, 1–4. Search in Google Scholar
[5] J. Cossey, Finite groups generated by subnormal T-subgroups, Glasg. Math. J. 37 (1995), no. 3, 363–371. 10.1017/S0017089500031645Search in Google Scholar
[6] K. Doerk and T. Hawkes, Finite Soluble Groups, De Gruyter Exp. Math. 4, Walter de Gruyter, Berlin, 1992. 10.1515/9783110870138Search in Google Scholar
[7] I. M. Isaacs, Semipermutable π-subgroups, Arch. Math. (Basel) 102 (2014), 1–6. 10.1007/s00013-013-0604-2Search in Google Scholar
[8] O. H. Kegel, Sylow-Gruppen und Subnormalteiler endlicher Gruppen, Math. Z. 78 (1962), 205–221. 10.1007/BF01195169Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- The role of the Rogers–Shephard inequality in the characterization of the difference body
- Test vectors for local periods
- The local Langlands conjecture for p-adic GSpin4, GSpin6, and their inner forms
- A family of irretractable square-free solutions of the Yang–Baxter equation
- On Sylow permutable subgroups of finite groups
- Using Steinberg algebras to study decomposability of Leavitt path algebras
- The ℓ∞-semi-norm on uniformly finite homology
- Minimal potential results for Schrödinger equations with Neumann boundary conditions
- On a unique continuation for Aharonov–Bohm magnetic Schrödinger equation
- Sup-norm bounds for Eisenstein series
-
Permutohedral structures on
- On the complete integrability of the periodic quantum Toda lattice
-
Fans, decision problems and generators of free abelian
- Standard cocycles: Variations on themes of C. Kassel’s and R. Wilson’s
- Crossed modules as maps between connected components of topological groups
- A local converse theorem for U(2,2)
- Probabilistic trace and Poisson summation formulae on locally compact abelian groups
