REPRESENTATIONS OF THE GUPTA-SIDKI GROUP

If p is an odd prime, then the Gupta-Sidki group G(p) is an infinite 2 -generated p-group. It is defined in a recursive manner as a particula r subgroup of the automorphism group of a regular tree of degree p. In this note, we make two observations concerning the irreducible repres entations of the group algebra K[G(p)] with K an algebraically closed field. First, when char K not equal p, we obtain a lower bound for the number of irreducible representations of any finite degree n. Second, when char K = p, we show that if K[G(p)] has one nonprincipal irreduc ible representation, then it has infinitely many. The proofs of these two results use similar techniques and eventually depend on the fact t hat the commutator subgroup H-p of G(p) has a normal subgroup of finit e index isomorphic to the direct product of p copies of H-p.