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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 b p |G|c/|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/ p |G|. Both of these bounds are best possible upper bounds for p(G), depending only on the order of G.

Probability that an element of a finite group has a square root