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

Volume 112 / 2008

M. S. Lucido, M. R. Pournaki Colloquium Mathematicum 112 (2008), 147-155 MSC: Primary 20A05, 20D60, 20P05; Secondary 05A15. DOI: 10.4064/cm112-1-7


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 $G$.


  • M. S. LucidoDipartimento di Matematica e Informatica
    Università di Udine
    Via delle Scienze 208
    I-33100 Udine, Italy
  • M. R. PournakiDepartment of Mathematical Sciences
    Sharif University of Technology
    P.O. Box 11155-9415
    Tehran, Iran
    School of Mathematics
    Institute for Studies in Theoretical Physics and Mathematics
    P.O. Box 19395-5746
    Tehran, Iran

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