Summary: On the probability of satisfying a word in a group
November 12, 2005
We show that for any finite group G and for any d there exists a word
w Fd 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, answering a
question of Alon Amit.
It also follows that there is no absolute bound in the Baumslag-Pride
theorem for the minimal index of a subgroup of a group with at least two
more generators than relators that can be mapped homomorphically onto
a nonabelian free group.
Let Fn denote the free group on n letters and let G be a group. For w Fn
we say that the n-tuple (g1, g2, . . . , gn) Gn
satisfies w if the substitution
w(g1, g2, . . . , gn) = 1. Our first result is the following.
Theorem 1 Let G be a finite group. Then for all n there exists a word w Fn
such that for all g1, g2, . . . , gn G, the tuple (g1, g2, . . . , gn) satisfies w if and