Number of involutions in a finite group [closed]

Question

Let $G$ be a finite group. Does it possible to determine number of involutions in it? If not, is there any bound for it?

Answer 1

There is a well-known precise formula for the number of involutions of a finite group in terms of character theory, which probably dates back to Frobenius and Schur. For each complex character, $\chi$, of $G$, the Frobenius-Schur indicator of $\chi$, denoted by $\nu(\chi)$, is $\frac{1}{|G|} \sum_{g \in G} \chi(g^{2}).$ We always have $\nu(\chi) \in \{0,-1,1 \}$ and we have $\nu(\chi) = 0 $ if $\chi$ is not real-valued, $\nu(\chi) = 1$ if $\chi$ is afforded by a real representation, $\nu(\chi) = -1$ otherwise.

The number of involutions of $G$ is $\sum_{\chi} \nu(\chi) \chi(1)$, where $\chi$ runs over all the non-trivial irreducible characters of $G$. This simple formula can be powerful. Note that, as expected, this gives no involutions when $G$ has odd order.

Using the Cauchy-Schwarz inequality, we obtain that the number of involutions of $G$ is at most $\sqrt{k-1} \sqrt{|G|-1}$, where $k$ is the number of conjugacy classes of $G$. which plays a role in deriving Brauer-Fowler type estimates.

For example, this gives $[G:C_{G}(t] < \sqrt{k|G|}$ for every involution $t \in G,$ which leads easily to $|G| < k|C_{G}(t)|^{2}$ for each involution $t \in G$ when $G$ has even order. (Later note: This bound is very close to being attained for all the simple groups ${\rm SL}(2,2^{n})$. For ${\rm SL}(2,2^{n})$ has order $(2^{n}-1)2^{n}(2^{n}+1)$ and has $(2^{n}+1)$ conjugacy classes, while each of its involutions has centralizer of order $2^{n}$). (Even later note: The number of involutions in ${\rm SL}(2,2^{n})$ is $2^{2n}-1$, while the $\sqrt{k-1}\sqrt{|G|-1}$ bound predicts for such groups that there are strictly fewer than $2^{n}\sqrt{2^{2n} -1}$ involutions, so the discrepancy is pretty small. The $\sqrt{k-1} \sqrt{|G|-1}$ bound reverts to the trivial $|G|-1$ bound when $G$ is Abelian, but to count involutions in an Abelian group is easy: the number of involutions in an Abelian group is $2^{r}-1$, where $r$ is the minimal number of generators of the Sylow $2$-subgroup).

These facts can be found in many character theory texts, (eg Isaacs).