Short two-variable identities for finite groups

In this paper, we consider finite groups G satisfying identities of the form x(e1)y(f1)x(e2)y(f2) ... x(er)y(fr) =1. We focus on identities with r small, Sigma(i) e(i) = Sigma(i) f(i) = 0, and all e(i), f(i) 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.