On Equational Theory of Group Conjugation

Given a group, there is a natural operation of conjugation, defined by x ∗ y = xyx−1. We study the variety generated by all G(∗), G a group. In particular, we are concerned about the question, whether this variety has finitely based equational theory. We study the equations satisfied by group conjugation. Strictly speaking, for a group G we put x∗y = xyx−1 for all x, y ∈ G and we ask, which equations hold in all groupoids G(∗), G a group. Particularly, does there exist a finite base of equations they satisfy? It is easy to see that every G(∗) is (1) idempotent (I), i.e. it satisfies the equation xx ≈ x, (2) left distributive (LD), i.e. it satisfies the equation x · yz ≈ xy · xz, (3) a left quasigroup, i.e. for every a ∈ G the left translation La : G→ G, x 7→ ax, is bijective. It turns out that LDI left quasigroups satisfy the same equations as all G(∗) do (it means, they generate the same variety). However, it wasn’t clear, whether they satisfy some additional equations, not following from LDI. In 1999, D. Larue found in [6] an infinite independent set of such equations of the form (xy · y)(x · z) ≈ (xy)(yx · z) (yxy · xy · y)(yx · z) ≈ (yxy · xy)(yyx · z) (yyxy · yxy · xy · y)(yyx · z) ≈ (yyxy · yxy · xy)(yyyx · z) etc. We use this result to get a conjecture which implies that the equations of conjugation are not finitely based. We also provide a couple of properties of the variety generated by all G(∗). 1991 Mathematics Subject Classification. Primary 20N02; secondary 20A99.