Let be a group and . Prove that if is a normal subgroup of then . Deduce that is the largest subgroup of in which is normal (i.e. the join of all subgroups for which is normal).
Note that if , then since is normal in . Hence , so that by a lemma to a previous exercise.
Now if we consider the set , note that and if then . Hence is -greatest among the subgroups of in which is normal.