Let $G$ be a finite group and denote by $S[G]$ the number of squares in $G$. The maximum, $S[G]=n$, is attained for a group of odd order $n$ since each element has a square root in that case. At the other extreme, $S[G]= 1$ when $G$ is an elementary abelian $2$-group because every element squares to the identity. Other examples include: $S[Sym(4)] = 12$ and $S[Sym(6)] = 270$ for symmetric groups, while both the dihedral and quaternion group of order $8$ have $S[G] = 2$.
- Can $S[G]$ be determined from the character table of $G$?
- Are the values of $S[G]$ known for the matrix groups $GL(2,q)$ and $GL(3,q)$?
- A table for $S[Sym(n)]$ is given in sequence A003483 at OEIS. For $n>3$, is it true that $S[Sym(n)]$ is divisible by all the primes less than or equal to $n$?