# 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”.