J. Aust. Math. Soc.  77 (2004), 185-189
On the orders of conjugacy classes in group algebras of p-groups

A. Bovdi
  University of Debrecen
  4010 Debrecen
  Hungary
  bodibela@math.klte.hu
L. G. Kovács
  Australian National University
  Canberra ACT 0200
  Australia
  kovacs@maths.anu.edu.au
and
S. Mihovski
  University of Plovdiv
  4000 Plovdiv
  Bulgaria
  mihovski@uni-plovdiv.bg


Abstract
Let  p be a prime, $\mathbb{F}$ a field of $p^n$ elements, and  G a finite  p-group. It is shown here that if  G has a quotient whose commutator subgroup is of order  p and whose centre has index $p^k$, then the group of normalized units in the group algebra $\mathbb{F} G$ has a conjugacy class of $p^{nk}$ elements. This was first proved by A. Bovdi and C. Polcino Milies for the case $k=2$; their argument is now generalized and simplified. It remains an intriguing question whether the cardinality of the smallest noncentral conjugacy class can always be recognized from this test.
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