On the orders of conjugacy classes in group algebras of p-groups

Let p be a prime, F a field of pⁿ 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 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.