Let be a subgroup of the group $G$ . Then $H$ is normal in $G$ if and only if it can be expressed as a union of conjugacy classes.

Proof

$H$ is normal in $G$ if and only if $g H g^{- 1} = H$ for all $g \in G$ ; equivalently, if and only if $g h g^{- 1} \in H$ for all $h \in H$ and $g \in G$ .

But if we fix $h \in H$ , then the statement that $g h g^{- 1} \in H$ for all $g \in G$ is equivalent to insisting that the conjugacy class of $h$ is contained in $H$ . Therefore $H$ is normal in $G$ if and only if, for all $h \in H$ , the conjugacy class of $h$ lies in $H$ .

If $H$ is normal, then it is clearly a union of conjugacy classes (namely $\cup_{h \in H} C_{h}$ , where $C_{h}$ is the conjugacy class of $h$ ).

Conversely, if $H$ is not normal, then there is some $h \in H$ such that the conjugacy class of $h$ is not wholly in $H$ ; so $H$ is not a union of conjugacy classes because it contains $h$ but not the entire conjugacy class of $h$ . (Here we have used that the conjugacy_classes_partition_the_group.)

Interpretation

A normal subgroup is one which is fixed under conjugation; the most natural (and, indeed, the smallest) objects which are fixed under conjugation are conjugacy classes; so this criterion tells us that to obtain a subgroup which is fixed under conjugation, it is necessary and sufficient to assemble these objects (the conjugacy classes), which are themselves the smallest objects which are fixed under conjugation, into a group.