A subset R of a finite group G is a square root of G if R2 = G. If R is a square root of G for which |R|2 = G, then R is referred to as a perfect square root of G. It can be shown using character theory that perfect square roots do not exist. The purpose of this paper is to work toward an elementary proof of this result.
Devos, Matthew; McAdams, David; and Rapoport, Rebecca, "Square Roots of Finite Groups - II" (1994). Mathematical Sciences Technical Reports (MSTR). 125.