This was asked at MSE but never answered.

Let $G$ be a finite group and denote by $sq(G)$ the number of squares in $G$ i.e. the number of elements in $G$ which possess a square root. For example, if $G$ is a group of odd order, then $sq(G) =|G|$, since each element has a square root, while at the opposite extreme, $sq(G)= 1$ when $G$ is an elementary abelian 2-group. For the symmetric groups, the values of $sq(\mathrm{Sym}(n))$ are listed at $\,$https://oeis.org/A003483 $\,$. Finally, we note that both the dihedral and quaternion groups of order 8 share the value $sq(G)=2$.

Questions:

1. Can $sq(G)$ be determined from information in the character table of $G$?

2. Is it true that $sq(\mathrm{Sym}(n))$ is divisible by every prime which is less than or equal to $n$ for $n>3$?

This was asked at MSE but never answered.

Let $G$ be a finite group and denote by $sq(G)$ the number of squares in $G$ i.e. the number of elements in $G$ which possess a square root. For example, if $G$ is a group of odd order, then $sq(G) =|G|$, since each element has a square root, while at the opposite extreme, $sq(G)= 1$ when $G$ is an elementary abelian 2-group. For the symmetric groups, the values of $sq(\mathrm{Sym}(n))$ are listed at $\,$https://oeis.org/A003483 $\,$. Finally, we note that both the dihedral and quaternion groups of order 8 share the value $sq(G)=2$.

Questions:

1. Can $sq(G)$ be determined from information in the character table of $G$?

2. Is it true that $sq(\mathrm{Sym}(n))$ is divisible by every prime which is less than or equal to $n$ for $n>3$?

This was asked at MSE but never answered.

Let G be a finite group and denote by$\,$ sq(G)$\,$ the number of squares in G$\,$, i.e. the number of elements in G which possess a square root. For example, if G is a group of odd order, $\;$ then$\qquad$ sq(G)$\, =\,$|G|$\;$since each element has a square root, while at the opposite extreme, sq(G)$\,$=$\,$1$\,$when G is an elementary abelian 2-group. For the symmetric groups, the values of sq(Sym(n)) are listed at $\,$https://oeis.org/A003483 $\,$. Finally, we note that both the dihedral and quaternion groups of order 8 share the value sq(G) = 2$\,$.

Questions: 1) Can $\,$sq(G)$\,$ be determined from information in the character table of G$\,$?

$\qquad$$\qquad$$\,$2) Is it true that $\,$sq(Sym(n))$\;$is divisible by every prime which is less than or equal to $\,$n$\,$ $\qquad$$\qquad$$\;$$\;$$\;$$\;$for $\,$n$\,$>$\,$3$\,$?

Thanks very much.

This was asked at MSE but never answered.

Let $G$ be a finite group and denote by $sq(G)$ the number of squares in $G$ i.e. the number of elements in $G$ which possess a square root. For example, if $G$ is a group of odd order, then $sq(G) =|G|$, since each element has a square root, while at the opposite extreme, $sq(G)= 1$ when $G$ is an elementary abelian 2-group. For the symmetric groups, the values of $sq(\mathrm{Sym}(n))$ are listed at $\,$https://oeis.org/A003483 $\,$. Finally, we note that both the dihedral and quaternion groups of order 8 share the value $sq(G)=2$.

Questions:

1. Can $sq(G)$ be determined from information in the character table of $G$?

2. Is it true that $sq(\mathrm{Sym}(n))$ is divisible by every prime which is less than or equal to $n$ for $n>3$?