Conjugacy classes of the group of units in group algebras of a finite p-groups
by
Adalbert Bovdi
University of Debrecen, Hungary

Let F_pⁿG be the group algebra of a finite p-group G over a field of pⁿ elements, and V(F_pⁿG) the subgroup of units of augmentation 1. Then V( F_pⁿG) is a finite p-group of order p^n(|G|-1). One of the hardest and most important problems for modular group algebras consists in describing the structure of V( F_pⁿG).

We know very little about the order conjugacy classes of V( F_pⁿG) for a finite p-group G. Rao and Sandling proved that p² divides the order of the conjugacy class C_a for any noncentral element a ∈ F_pG. We improve this result is the following way:

THEOREM. Let G be a finite p-group and F_pⁿ be a finite field of pⁿ elements. Then p²ⁿ divides the order |C_a| for every noncentral a ∈ F_pⁿG.

It is still open when exist conjugacy classes of V( F_pⁿG) of order p^kn. It is well-know fact that for odd p in finite p-group G there exists a normal subgroup H such that G/H has the commutator subgroup of order p and G/H is the following central product:

G/H=(G/H)0Y(G/H)1Y(G/H)2Y...Y(G/H)s, \tag1

where (G/H)_i∩(G/H)_j=〈cH〉 and the factors are the following groups: (G/H)₀ is cyclic and for i > 1 the group (G/H)_i=〈a_i, b_i〉 is nonabelian group of order p³ and of exponent p.

QUESTION. Suppose that H=1 and G has a decomposition (1). Is it true that p^2sn divide the order every conjugacy classes of V( F_pⁿG) for the odd p?

THEOREM. Let G be a nonabelian finite p-group of odd order and G has a factor group G/H with the commutator subgroup of order p such that (1) is satisfied. Let

y= s Õ i=1  (bi-1)p-1(ai-1)p-1(c-1)p-2 ^ H   .

The conjugacy class C_y of V(F_pⁿG) has the order |F_pⁿ|^2s.

It follows the result of the author and Polcino Milies in the case s=1 that V( F_pⁿG) contains a conjugacy class of order p²ⁿ.

Date received: June 28, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafe-21.