# Number of involutions in a finite group modulo $4$

It is a theorem of Alperin, Feit and Thompson [Isaacs Character Theory book (4.9)] that if $G$ is a $2$-group in which the number of involutions is congruent to $1$ modulo $4$, then $G$ is cyclic or $|G:G'|=4$. In the latter case $G$ is dihedral, semidihedral or generalized quaternion.

Are there any results of a similar nature which hold for all finite groups? I want a result that says that in a finite group the number of involutions, modulo $4$, influences the structure of a Sylow $2$-subgroup of the group.

EDIT: Actually, there is an older reference which will give the result below. See

Herzog, Marcel Counting group elements of order $p$ modulo $p^2$. Proc. Amer. Math. Soc. 66 (1977), no. 2, 247–250.

where you will also find references to other related papers.

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The following paper seems to give what you are looking for.

Murai, Masafumi On the number of p-subgroups of a finite group. J. Math. Kyoto Univ. 42 (2002), no. 1, 161–174. link

Let $G$ be a finite group and fix a prime $p$. Denote by $s_e(G)$ the number of subgroups of order $p^e$ in $G$. If $p^e$ divides the order of $G$, a classical result of Frobenius states that $s_e(G) \equiv 1 \mod{p}$. The paper of Murai above studies how $s_e(G) \mod{p^2}$ depends on the structure of a Sylow $p$-subgroup of $G$.

For $p = 2$, Theorem D in the paper implies the following:

Let $G$ be a finite group with a Sylow $2$-subgroup $P$ of order $2^n$ ($n \geq 2$). If $P$ is cyclic, dihedral, generalized quaternion, or semidihedral, then $s_1(G) \equiv 1 \mod{4}$. Otherwise $s_1(G) \equiv 3 \mod{4}$.

Note that in this case $s_1(G)$ is the number of involutions in $G$.