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Large centralizers in finite solvable groups

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Abstract

In 1955 R. Brauer and K. A. Fowler showed that ifG is a group of even order >2, and the order |Z(G)| of the center ofG is odd, then there exists a strongly real) elementx∈G−Z whose centralizer satisfies|C G(x)|>|G|1/3. In Theorem 1 we show that every non-abeliansolvable groupG contains an elementx∈G−Z such that|C G(x)|>[G:G′∩Z]1/2 (and thus|C G(x)|>|G|1/3). We also note that if non-abelianG is either metabelian, nilpotent or (more generally) supersolvable, or anA-group, or any Frobenius group, then|C G(x)|>|G|1/2 for somex∈G−Z. In Theorem 2 we prove that every non-abelian groupG of orderp mqn (p, q primes) contains a proper centralizer of order >|G|1/2. Finally, in Theorem 3 we show that theaverage |C(x)|, x∈G, is ≧c|G| 1/3 for metabelian groups, wherec is constant and the exponent 1/3 is best possible.

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  1. Mathematics Department, University of Hawaii-Manoa, 96822, Honolulu, HI, USA

    Edward A. Bertram

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  1. Edward A. Bertram
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Bertram, E.A. Large centralizers in finite solvable groups. Israel J. Math. 47, 335–344 (1984). https://doi.org/10.1007/BF02760607

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  • DOI: https://doi.org/10.1007/BF02760607

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