Subgroups generated by small classes in finite groups
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- by I. M. Isaacs
- Proc. Amer. Math. Soc. 136 (2008), 2299-2301
- DOI: https://doi.org/10.1090/S0002-9939-08-09263-0
- Published electronically: March 14, 2008
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Abstract:
Let $M(G)$ be the subgroup of $G$ generated by all elements that lie in conjugacy classes of the two smallest sizes. Avinoam Mann showed that if $G$ is nilpotent, then $M(G)$ has nilpotence class at most $3$. Using a slight variation on Mann’s methods, we obtain results that do not require us to assume that $G$ is nilpotent. We show that if $G$ is supersolvable, then $M(G)$ is nilpotent with class at most $3$, and in general, the Fitting subgroup of $M(G)$ has class at most $4$.References
- Kenta Ishikawa, On finite $p$-groups which have only two conjugacy lengths, Israel J. Math. 129 (2002), 119–123. MR 1910937, DOI 10.1007/BF02773158
- Noboru Itô, On finite groups with given conjugate types. I, Nagoya Math. J. 6 (1953), 17–28. MR 61597
- Avinoam Mann, Elements of minimal breadth in finite $p$-groups and Lie algebras, J. Aust. Math. Soc. 81 (2006), no. 2, 209–214. MR 2267792, DOI 10.1017/S1446788700015834
Bibliographic Information
- I. M. Isaacs
- Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706
- Email: isaacs@math.wisc.edu
- Received by editor(s): March 26, 2007
- Published electronically: March 14, 2008
- Communicated by: Jonathan I. Hall
- © Copyright 2008 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 136 (2008), 2299-2301
- MSC (2000): Primary 20D25
- DOI: https://doi.org/10.1090/S0002-9939-08-09263-0
- MathSciNet review: 2390495