Semigroup identities and proofs

Tamura proved that for any semigroup word U(x, y), if every group satisfying an identity of the form yx similar to xU(x, y)y is abelian, then so is every semigroup that satisfies that identity. Because a group has an identity element and the cancellation property, it is easier to show that a group is abelian than that a semigroup is. If we know that it is, then there must be a sequence of substitutions using xU(x, y)y similar to yx that transforms xy to yx. We examine such sequences and propose finding them as a challenge to proof by computer. Also, every model of y similar to xU(x, y)x is a group. This raises a similar challenge, which we explore in the special case y similar to x (m) y (p) x (n) . In addition, we determine the free model with two generators of some of these identities. In particular, we find that the free model for y similar to x (2) yx (2) has order 32 and is the product of D (4) (the symmetries of a square), C (2), and C (4), and point out relations between such identities and Burnside's Problem concerning models of x (n) similar to y (n) . We also examine several identities not related to groups.