Probability that an element of a finite group has a square root
Volume 112 / 2008
Colloquium Mathematicum 112 (2008), 147-155
MSC: Primary 20A05, 20D60, 20P05; Secondary 05A15.
DOI: 10.4064/cm112-1-7
Abstract
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 $G$.
