Subgroup is normal if and only if it is a union of conjugacy classes

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.

Parents:

  • Normal subgroup

    Nor­mal sub­groups are sub­groups which are in some sense “the same from all points of view”.