Derived subgroups and centers of capable groups
HTML articles powered by AMS MathViewer
- by I. M. Isaacs PDF
- Proc. Amer. Math. Soc. 129 (2001), 2853-2859 Request permission
Abstract:
A group $G$ is said to be capable if it is isomorphic to the central factor group $H/\mathbf {Z}(H)$ for some group $H$. It is shown in this paper that if $G$ is finite and capable, then the index of the center $\mathbf {Z}(G)$ in $G$ is bounded above by some function of the order of the derived subgroup $G’$. If $G’$ is cyclic and its elements of order $4$ are central, then, in fact, $|G:\mathbf {Z}(G)| \le |G’|^{2}$.References
- F. Rudolf Beyl and Jürgen Tappe, Group extensions, representations, and the Schur multiplicator, Lecture Notes in Mathematics, vol. 958, Springer-Verlag, Berlin-New York, 1982. MR 681287
- Ying Cheng, On finite $p$-groups with cyclic commutator subgroup, Arch. Math. (Basel) 39 (1982), no. 4, 295–298. MR 684396, DOI 10.1007/BF01899434
- Hermann Heineken, Nilpotent groups of class two that can appear as central quotient groups, Rend. Sem. Mat. Univ. Padova 84 (1990), 241–248 (1991). MR 1101296
- Hermann Heineken and Daniela Nikolova, Class two nilpotent capable groups, Bull. Austral. Math. Soc. 54 (1996), no. 2, 347–352. MR 1411544, DOI 10.1017/S0004972700017809
- I. M. Isaacs, Automorphisms fixing elements of prime order in finite groups, Arch. Math. (Basel) 68 (1997), no. 5, 359–366. MR 1441634, DOI 10.1007/s000130050068
- H. Wielandt, Eine Verallgemeinerung der invarianten Untergruppen, Math. Zeit. 45 (1939) 209–244.
Additional 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): December 20, 1999
- Received by editor(s) in revised form: February 22, 2000
- Published electronically: February 22, 2001
- Additional Notes: This paper was written with the partial support of the U.S. National Security Agency.
- Communicated by: Stephen D. Smith
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 129 (2001), 2853-2859
- MSC (2000): Primary 20D99
- DOI: https://doi.org/10.1090/S0002-9939-01-05888-9
- MathSciNet review: 1840087