on group elements having square roots

On group Elements Having Square Roots

Elements with Square Roots in Finite Groups

In this paper, we study the probability that a randomly chosen element in a finite group has a square root, in particular the simple groups of Lie type of rank 1, the sporadic finite simple groups and the alternating groups.

Square roots of Thompson’s group F

We study two–generated subgroups x f , gy Homeo pIq such that x f 2, g2y is isomorphic to Thompson’s group F. We show that this class contains examples with nonabelian free subgroups, examples which do not admit faithful actions by C diffeomorphisms on 1–manifolds, and examples which are not finitely presented. We thus answer questions due to M. Brin.

Two–chains and square roots of Thompson’s group F

We study two–generated subgroups x f , gy Homeo pIq such that x f 2, g2y is isomorphic to Thompson’s group F, and such that the supports of f and g form a chain of two intervals. We show that this class contains uncountably many isomorphism types. These include examples with nonabelian free subgroups, examples which do not admit faithful actions by C2 diffeomorphisms on 1–manifolds, examples ...

On the denesting of nested square roots

We present the basic theory of denesting nested square roots, from an elementary point of view, suitable for lower-level coursework. Necessary and sufficient conditions are given for direct denesting, where the nested expression is rewritten as a sum of square roots of rational numbers, and for indirect denesting, where the nested expression is rewritten as a sum of fourth-order roots of ration...

On Square Roots of Normal Operators1

It is clear that if A7 possesses the spectral resolution N = jzdK(z), then any operator of the form A =Jzll2dK(z), where, for the value of z1'2, the choice of the branch of the function may depend on z, is a solution of (1). Moreover, all such operators are even normal. Of course, equation (1) may have other, nonnormal, solutions A. The object of this note is to point out a simple condition to ...