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Let $G$ be a finite group of even order. We
give some bounds for the probability ${\rm p}(G)$ that a randomly
chosen element in $G$ has a square root. In particular, we prove
that ${\rm p}(G) \leq 1-{\lfloor \sqrt|G|\rfloor/|G|}$. Moreover,
we show that if the Sylow 2-subgroup of $G$ is not a proper normal
elementary abelian subgroup of $G$, then ${\rm p}(G) \le
1-1/\sqrt|G|$. Both of these bounds are best possible upper bounds
for ${\rm p}(G)$, depending only on the order of $G$.
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