PROBABILITY THAT AN ELEMENT OF A FINITE GROUP HAS A SQUARE ROOT BY

Abstract. Let G be a finite group of even order. We give some bounds for the probability p(G) that a randomly chosen element in G has a square root. In particular, we prove that p(G) ≤ 1 − ⌊ |G|⌋/|G|. Moreover, we show that if the Sylow 2-subgroup of G is not a proper normal elementary abelian subgroup of G, then p(G) ≤ 1 − 1/ |G|. Both of these bounds are best possible upper bounds for p(G), depending only on the order of G. 1. Introduction. Let G be a finite group and let g ∈ G. If there exists an element h ∈ G for which g = h2, then we say that g has a square root. Clearly, g may have one or more square roots, or it may have none. Let G2 be the set of all elements of G which have at least one square root, i.e., G2 = {g ∈ G | there exists h ∈ G such that g = h2}