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

https://arbital.com/p/subgroup_normal_iff_union_of_conjugacy_classes

by Patrick Stevens Jun 17 2016

A useful way to think about normal subgroups, which meshes with their "closed under conjugation" interpretation.


Let be a subgroup of the Group . Then is normal in if and only if it can be expressed as a union of conjugacy classes.

Proof

is normal in if and only if for all ; equivalently, if and only if for all and .

But if we fix , then the statement that for all is equivalent to insisting that the conjugacy class of is contained in . Therefore is normal in if and only if, for all , the conjugacy class of lies in .

If is normal, then it is clearly a union of conjugacy classes (namely , where is the conjugacy class of ).

Conversely, if is not normal, then there is some such that the conjugacy class of is not wholly in ; so is not a union of conjugacy classes because it contains but not the entire conjugacy class of . (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.