A characterization of finite soluble groups

Let G be a finite soluble group of order m and let ω(xi, ⋯, xn) be a group word. Then the probability that ω(g1, ⋯, gn) = 1 (where (g1, mellip;, gn) is a random n-tuple in G)isatleast p-(m-t), where p is the largest prime divisor of m and t is the number of distinct primes dividing m. This contrasts...