On Finite Groups with Some Conditions on Subsets

Let n be a positive integer. We denote by W*(n) the class of groups G such that, for every subset X of G of cardinality n + 1, there exist a positive integer k, and a subset X0 ⊆ X, with 2 ≤ |X0| ≤ n + 1 and a function f : {0; 1; 2; : : : ; k} rightarrow X0, with f(0) neq f(1) and non-zero integers t0; t1; : : : ; tk such that [x^t0 0 ; x^t1 1 ; : : : ; x^tk k ] = 1, where xi := f(i), i = 0; : : : ; k, and xj ∈ H whenever x ^tj j ∈ H, for some subgroup H neq x^tj j of G. If the integer k is xed for every subset X we obtain the class W* k (n). If one always has ti = 1, i = 0; 1; : : : ; k, and x0 = x; xi = y, i = 1; 2; : : : ; k, one obtains the class Ek(n). In this paper, we prove that (1) A nite semi-simple group has the property W* k (5), for some k, if and only if G asymptotically equal to A5 or S5, (3) A nite non-nilpotent group has the property W* k (3), for some k, if and only if G=Z*(G) asymptotically equal to S3, where Z*(G) is the hypercenter of G, (2) A nite semi-simple group has the property Ek(16), for some k, if and only if G asymptotically equal to A5, where An and Sn denote the alternating and symmetric groups of degree n respectively.