Let $~$H$~$ 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 $~$gHg^{-1} = H$~$ for all $~$g \in G$~$; equivalently, if and only if $~$ghg^{-1} \in H$~$ for all $~$h \in H$~$ and $~$g \in G$~$.

But if we fix $~$h \in H$~$, then the statement that $~$ghg^{-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.