J. Aust. Math. Soc.  78 (2005), 291-295
Elements of prime power order and their conjugacy classes in finite groups

László Héthelyi
  Department of Algebra
  Technical University of Budapest
  H-1521 Budapest
  Hungary
  hethelyi@math.bme.hu
and
Burkhard Külshammer
  Mathematical Institute
  University of Jena
  D-07737 Jena
  Germany
  kuelshammer@uni-jena.de


Abstract
We show that, for any positive integer $k$, there are only finitely many finite groups, up to isomorphism, with exactly $k$ conjugacy classes of elements of prime power order. This generalizes a result of E. Landau from 1903. The proof of our generalization makes use of the classification of finite simple groups.
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