On the intersection of a class of maximal subgroups of a finite group
HTML articles powered by AMS MathViewer
- by Xiu Yun Guo PDF
- Proc. Amer. Math. Soc. 106 (1989), 329-332 Request permission
Abstract:
Let $G$ be a finite group and $\pi$ a set of primes. We consider the family of subgroups of $G:\mathcal {F} = \{ M:M < \cdot G,{[G:M]_\pi } = 1,[G:M]$ is composite} and denote ${S_\pi }(G) = \bigcap \left \{ M: M \in \mathcal {F} \right \}$ if $\mathcal {F}$ is non-empty, otherwise ${S_\pi }(G) = G$. The purpose of this note is to prove Theorem. Let $G$ be a $\pi$-solvable group. Then ${S_\pi }(G)$ has the following properties: (1) ${S_\pi }(G)/{O_\pi }(G)$ is supersolvable. (2) ${S_\pi }({S_\pi }(G)) = {S_\pi }(G)$. (3) $G/{O_\pi }(G)$ is supersolvable if and only if ${S_\pi }(G) = G$.References
- W. E. Deskins, On maximal subgroups, Proc. Sympos. Pure Math., Vol. 1, American Mathematical Society, Providence, R.I., 1959, pp. 100–104. MR 0125157
- Wolfgang Gaschütz, Über die $\Phi$-Untergruppe endlicher Gruppen, Math. Z. 58 (1953), 160–170 (German). MR 57873, DOI 10.1007/BF01174137
- John S. Rose, The influence on a finite group of its proper abnormal structure, J. London Math. Soc. 40 (1965), 348–361. MR 174638, DOI 10.1112/jlms/s1-40.1.348
- Prabir Bhattacharya and N. P. Mukherjee, On the intersection of a class of maximal subgroups of a finite group. II, J. Pure Appl. Algebra 42 (1986), no. 2, 117–124. MR 857561, DOI 10.1016/0022-4049(86)90074-5
- Henry G. Bray, W. E. Deskins, David Johnson, John F. Humphreys, B. M. Puttaswamaiah, Paul Venzke, and Gary L. Walls, Between nilpotent and solvable, Polygonal Publ. House, Washington, N.J., 1982. Edited and with a preface by Michael Weinstein. MR 655785
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 106 (1989), 329-332
- MSC: Primary 20D20
- DOI: https://doi.org/10.1090/S0002-9939-1989-0999757-6
- MathSciNet review: 999757