Illinois Journal of Mathematics

On $\mathcal{M}$-permutable sylow subgroups of finite groups

Long Miao and Wolfgang Lempken

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A $p$-subgroup $P\neq 1$ of $G$ is called $\mathcal{M}$-permutable in $G$ if there exists a set $\mathcal{ M}_d(P)=\{P_1,\ldots,P_d\}$ of maximal subgroup $P_i$ of $P$ and a subgroup $B$ of $G$ such that: (1) $\bigcap_{i=1}^{d}{P_i}=\Phi(P)$ and $|P : \Phi(P)|=p^d$; (2) $G=PB$ and $P_iB=BP_i \lt G$ for any $P_i$ of $\mathcal{M}_d(P)$. In this paper, we investigate the influence of $\mathcal M$-permutability of Sylow subgroups in finite groups. Some new results about supersolvable groups and formations are obtained.

Article information

Illinois J. Math., Volume 53, Number 4 (2009), 1095-1107.

First available in Project Euclid: 22 November 2010

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Zentralblatt MATH identifier

Primary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $\pi$-length, ranks [See also 20F17] 20D20: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure


Miao, Long; Lempken, Wolfgang. On $\mathcal{M}$-permutable sylow subgroups of finite groups. Illinois J. Math. 53 (2009), no. 4, 1095--1107. doi:10.1215/ijm/1290435341.

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  • Z. Arad and M. B. Ward, New criteria for the solvability of finite groups, J. Algebra 77 (1982), 234–246.
  • A. Ballester-Bolinches, Y. Wang and X. Guo, $C$-supplemented subgroups of finite groups, Glasgow Math. J. 42 (2000), 383–389.
  • D. Gorenstein, Finite groups, Harper & Row, New York, 1968.
  • W. Guo, The theory of classes of groups, Kluwer Academic Publishers, Dordrecht, 2000.
  • P. Hall, A characteristic property of soluble groups, J. London Math. Soc. 12 (1937), 188–200.
  • B. Huppert, Endliche gruppen I, Springer, Berlin, 1967.
  • B. Huppert and N. Blackburn, Finite groups III, Springer, Berlin, 1982.
  • S. Li and X. He, On normally embedded subgroups of prime power order in finite groups, Comm. Algebra 36 (2008), 2333–2340.
  • L. Miao and W. Lempken, On $\mathcal{M}$-supplemented subgroups of finite groups, J. Group Theory 12 (2009), 271–289.
  • L. Miao, On p-nilpotency of finite groups, Bull. Braz. Math. Soc. 38 (2007), 585–594.
  • D. J. Robinson, A course in the theory of groups, Springer, Berlin, 1993.
  • A. N. Skiba, A note on $c$-normal subgroups of finite groups, Algebra Discrete Math. 3 (2005), 85–95.
  • S. Srinivasan, Two sufficient conditions for supersolvability of finite groups, Israel J. Math. 35 (1980), 210–214.
  • J. Tate, Nilpotent quotient groups, Topology 3 (1964), 109–111.
  • J. G. Thompson, Normal $p$-complement for finite groups, J. Algebra 1 (1964), 43–46.
  • Y. Wang, Finite groups with some subgroups of Sylow subgroups c-supplemented, J. Algebra 224 (2000), 467–478.\goodbreak
  • Y. Wang, H. Wei and Y. Li, A generalization of Kramer's theorem and its applications, Bull. Austral. Math. Soc. 65 (2002), 467–475.