Let be a group. Prove that the join of any nonempty collection of normal subgroups of is normal.
Lemma: Let be a group, , and a nonempty collection of subgroups of . Then . Proof: Let . Then for some . Thus there exists with , so that . Hence . Let . Then there exists with , so that for some . But , so that .
Recall that the join of is . Now let ; we saw in a previous exercise that . Hence is normal in .