In 2020, Mousavi proved that a finite group $G$ with a generalized quaternion Sylow $2$-subgroup $S$ is $2$-nilpotent if $3\nmid |G|$ or if $G$ is solvable and $|S|>16$. In this note, we generalized the result of Mousavi and provided a simpler proof. In detail, we showed that a finite group with a generalized quaternion Sylow $2$-subgroup is $2$-nilpotent if, and only if, it is $\mathrm{S}{\mathrm{L}}_2(3)$-free and that, for a finite group with a generalized quaterion Sylow $2$-subgroup of order strictly greater than $16$, the properties of being $2$-nilpotent, solvable, and $2$-constrained are equivalent.

Let $G$ be a group and $S$ be a subset of $G.$ We say that $S$ normally generates $G$ if $G$ is the normal closure of $S$ in $G.$ In this situation, every element $g\in G$ can be written as a product of conjugates of elements of $S$ and their inverses. If $S\subseteq G$ normally generates $G,$ then the length $\Vert g{\Vert }_S\in \mathbb{N}$ of $g\in G$ with respect to $S$ is the shortest possible length of a word in ${\text{Conj}}_G(S^{\pm 1}):=\{h^{-1}sh|h\in G,s\in S\text{or}s{}^{-1}\in S\}$ expressing $g.$ We write $\Vert G{\Vert }_S=\text{sup}\{\Vert g{\Vert }_S|g\in G\}$ for any normally generating subset $S$ of $G.$ The conjugacy diameter of any group $G$ is $\mathrm{\Delta }(G):=sup\{\Vert G{\Vert }_S|S\text{is a finite normally generating subset of }G\}.$ We say that $G$ is uniformly bounded if $\mathrm{\Delta }(G)<\mathrm{\infty }.$ This concept is a strengthening of boundedness. Motivated by previously known results approximating $\mathrm{\Delta }(G)$ for any algebraic group $G,$ we find the exact values of the conjugacy diameters of the direct product of finitely many copies of $S{L}_2(\mathbb{C})$ and the direct product of finitely many copies of $PS{L}_2(\mathbb{C}).$ We also prove that if $G_1,\dots ,G_n$ be quasisimple groups such that $G_i$ is uniformly bounded for each $i\in \{1,\dots ,n\},$ then $G_1\times \cdots \times G_n$ is uniformly bounded. This is also a generalization of some previously known results in the literature.

Let $G$ be a group. A subset $S$ of $G$ is said to normally generate $G$ if the normal closure of $S$ in $G$ is equal to $G$ itself. This means that every element of $G$ can be represented as a product of conjugates of elements of $S$ and their inverses. Given an element $g$ of $G$ and a normally generating set $S,$ we define the length of $g$ with respect to $S$ as the smallest number of conjugates of elements of $S$ or their inverses needed to express $g$ as a product. Then, for each such $S$, the diameter of $G$ with respect to $S$ is defined as the supremum of the lengths of elements of $G$ with respect to $S.$ The conjugacy diameter of $G$ is the supremum of all diameters of $G$ over all finite normally generating subsets. It measures how efficiently $G$ is normally generated by its finite normally generating subsets.

In this paper, we found the conjugacy diameters of finite dihedral groups. It is worth noting that the conjugacy diameters of other families, such as semidihedral $2$-groups, generalized quaternion groups, and modular $p$-groups, have already been investigated.