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”.