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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 $S[G] = 2$.

Questions:

  1. Can $S[G]$ be determined from the character table of $G$?
  2. Are the values of $S[G]$ known for the matrix groups $GL(2,q)$ and $GL(3,q)$?
  3. 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 $n$?

Thanks

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    $\begingroup$ It might be good to post these as separate questions; I think I could probably answer question 3 if I thought about it, but am less sure about the others - you'd be more likely to get an answer for each one individually if you separated them (and you could put links to the others in each one) $\endgroup$
    – Milo Brandt
    Jan 24, 2015 at 18:13
  • $\begingroup$ Crossposted to MO (at least 1 and 3): mathoverflow.net/questions/221157/… $\endgroup$
    – YCor
    Oct 18, 2015 at 9:57

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