Abstract

We show that for any finite group G and for any d there exists a word w ∈ F_d such that a d-tuple in G satisfies w if and only if it generates a solvable subgroup. As a corollary, the probability that a word is satisfied in a fixed non-solvable group can be made arbitrarily small; this answers a question of Alon Amit.

It also follows that there is no absolute bound in the Baumslag–Pride theorem for the minimal index in a group with at least two more generators than relators of a subgroup that can be mapped homomorphically onto a non-abelian free group.