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Finite groups with large centralizers

  • Edward A. Bertram (a1) and Marcel Herzog (a2)

Abstract

It is known that a finite non-abelian group G has a proper centralizer of order if, for example, |G| is even and |Z(G)| is odd, or whenever G is solvable. Often the exponent can be improved to , for example when G is supersolvable, or metabelian, or |G = pαqβ. Here we show more generally that this improvement is possible in many situations where G is factorizable into the product of two subgroups. In particular, much more evidence is presented to support the conjecture that some proper centralizer has order whenever G is a finite non-abelian solvable group.

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Copyright

References

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[1]Baer, R., “Partitionen endlicher Gruppen”, Math. Zeit. 75 (1961) 333372.
[2]Bertram, E. A., “Large centralizers in finite solvable groups”, Israel J. Math. 47 (1984), 335344.
[3]Gorenstein, D., “Finite Groups”, (Harper and Row, New York, 1968).
[4]Huppert, B., “Endlicher Gruppen”, Vol. I, (springer-Verlag, Berlin, New York, 1967).
[5]Rocke, D. M., “Groups with abelian centralizers”, Unpub. Part of Ph.D. Diss., Univ. of Illinois, 1972.
[6]Schmidt, R., “Zentralisatorverbände endlicher Gruppen”, Sem. Math. De Univ. Padova 44 (1970), 97111.
[7]Suzuki, M., “Group Theory I”, Springer-Verlag, Berlin-Heidelberg-New York, 1982.
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Finite groups with large centralizers

  • Edward A. Bertram (a1) and Marcel Herzog (a2)

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