Proving a property of normal subgroups.

Question

Let $G$ be a group and let $G_1,G_2$ be subgroups of $G$ with $G_1\cap G_2= \{e\}$. Show that $G_1$ and $G_1$ commute if and only if they are normal subgroups. With $G_1G_2 $ defined as {$xy \in G : x\in G_1 , y\in G_2$}

1) I think they both need not be normal subgroup. If even one of them is normal they will definitely commute. (Tell me if I am wrong).

2) But I don't know how to prove they don't commute if both are not normal. Pls help me and thanxx for the help.

Answer 1

Your claim is false. Take $G=S_n (n\geq 4),G_1=\lbrace id,(12) \rbrace, G_2=\lbrace id,(34) \rbrace$.

Then $G_1G_2=G_2G_1$ but neither $G_1$ nor $G_2$ is normal.