Maximum conjugacy class size

Definition

For $G$ a group, the maximum conjugacy class size is the largest size of all the conjugacy classes in the group. If the largest conjugacy class is (countably, uncountably, etc.) infinite then we say this is (countably, uncountably, etc.) infinite.

Examples

• The maximum conjugacy class size of any abelian group $G$ is $1$, since $hgh^{-1}=g$ for all $g,h\in G$.

• The maximum conjugacy class size of the quaternion group is $2$, since the conjugacy classes are $\{1\}$, $\{-1\}$, $\{i,-i\}$, $\{j,-j\}$, $\{k,-k\}$.

• The maximum conjugacy class size of the dihedral group of order $2n$ is $n$ if $n$ is odd, $n/2$ if $n$ is even.