Short two-variable identities for finite groups

Abstract

In this paper, we consider finite groups G satisfying identities of the form x1y1x2y2 . . . xryr = 1 . We focus on identities with r small, ∑ i ei = ∑ i fi = 0, and all ei, fi coprime to the order of G. We show that for r = 2, 3 and 5, G must be nilpotent. We also classify for r = 4, 6 and 7, the special identities which can hold in non-nilpotent groups. Finally, we show that for r < 30, the group G must be solvable.