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.