Normal Subgroups and Cosets
For , the subgroup is normal in exactly when for every .
Lemma 7.3
Suppose . Then iff
for all .
Proof
If , then gives for every , so . Applying the same argument to gives , hence .
Conversely, if for every , then for , so for some . Therefore , and .
Related
Stated in
- Lemma 7.3§7.1 Normal Subgroups