Numerical bounds for the exterior degree of finite simple groups

The exterior degree d<^>(G) of a finite group G is the probability that a pair (x, y) of elements x, y chosen uniformly at random in G satisfies x <^> y = 1(<^>), where <^> is the operator of nonabelian exterior square G <^> G and 1(<^>) neutral element in G <^> G. The probability d(G) that two elements of G commute is related to d<^>(G). Of course, d(G) = 1 iff G is abelian, but d<^>(G) = 1 iff G is cyclic, so we detect cyclic groups when d<^>(G) is close to 1. We present new numerical results for the exterior degree of finite simple groups.