Proof normal divider of two sets?

Question

Let be $ U $ and $ V$ normal dividers of G with $ U \cap V = \{ e\} $.

How can I show that $W= \{ a * b | a \in U, b \in V \} $ is also a normal divider of $ G$ and that $W$ isomorphic to $ U \times V$?

Can I use : $ U $ normal divider : $g^{-1} *a *g \in U $ with $a \in U, g \in G$ $ V $ normal divider : $g^{-1} *b *g \in V $ with $b \in V, g \in G$ Would be very grateful if someone gave me a hint, because I'm stuck in a loop

Answer 1

Start by proving that elements of $U$ commute with elements of $V$. Take $a\in U$ and $b\in V$ and prove that: $$[a,b]=aba^{-1}b^{-1} = 1$$ (Hint: put parentheses in two different ways in that expression).

This will show that $W\simeq U\times V$.

Next, you will have to prove that $W$ is a subgroup of $G$ and that $W\lhd G$.