A characterization of finite soluble groups

Let G be a finite soluble group of order m and let w(x₁, ...,x_n) be a group word. Then the probability that w(g₁, ..., g_n)= 1 (where (g₁, ..., g_n) is a random n-tuple in G) is at leastp^–(m–t), where p is the largest prime divisor ofm and t is the number of distinct primes dividing m. This contrastswith the case of a non-soluble group G, for which Abérthas shown that the corresponding probability can take arbitrarilysmall positive values as n .