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Derived subgroups and centers of capable groups

Author(s): I. M. Isaacs
Journal: Proc. Amer. Math. Soc. 129 (2001), 2853-2859.
MSC (2000): Primary 20D99
Posted: February 22, 2001
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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, $\vert G:\mathbf{Z}(G)\vert \le \vert G'\vert^{2}$.

References:

1.
F. R. Beyl and J. Tappe, Extensions, Representations and the Schur Multiplicator, Lecture Notes in Mathematics 958, Berlin, Heidelberg, New York, 1989. MR 84f:20002

2.
Y. Cheng, On finite $p$-groups with cyclic commutator subgroup, Arch. Math. 39 (1982) 295-298. MR 84c:20028

3.
H. Heineken, Nilpotent groups of class two that can appear as central quotient groups, Rend. Sem. Mat. Univ. Padova 84, (1990) 241-248. MR 92c:20068

4.
H. Heineken and D. Nikolova, Class two nilpotent capable groups, Bull. Austral. Math. Soc. 54 (1996) 347-352. MR 97m:20043

5.
I. M. Isaacs, Automorphisms fixing elements of prime order in finite groups, Arch. Math. 68 (1997) 357-366. MR 98e:20030

6.
H. Wielandt, Eine Verallgemeinerung der invarianten Untergruppen, Math. Zeit. 45 (1939) 209-244.

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Additional Information:

I. M. Isaacs
Affiliation: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, Wisconsin 53706
Email: isaacs@math.wisc.edu

DOI: 10.1090/S0002-9939-01-05888-9
PII: S 0002-9939(01)05888-9
Received by editor(s): December 20, 1999
Received by editor(s) in revised form: February 22, 2000
Posted: 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 of article: Copyright 2001, American Mathematical Society

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