A characterization of finite soluble groups

Let G be a finite soluble group of order m and let w (x(1),..., x(n)) be a group word. Then the probability that w (g(1),..., g(n)) = 1 (where (g(1),..., g(n)) is a random n-tuple in G) is at least p(-(m-t)), where p is the largest prime divisor of m and t is the number of distinct primes dividing m. This contrasts with the case of a non-soluble group G, for which Abert has shown that the corresponding probability can take arbitrarily small positive values as n -> infinity.