# How many squares in a finite group?

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

• 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) Jan 24, 2015 at 18:13
• Crossposted to MO (at least 1 and 3): mathoverflow.net/questions/221157/…
– YCor
Oct 18, 2015 at 9:57