My course notes contain the proposition,
Let $G$ be a group and $G_1 , G_2$ normal subgroups of $G$ such that $G = G_1 G_2$ and $G_1 \cap G_2 = \{e\}$. Then $G \cong G_1 \times G_2$.
An example is given using $\mathbb{Z}_6 \cong \langle 2 \rangle \times \langle 3 \rangle$.
I think I can see that $\mathbb{Z}_6 \cong \langle 2 \rangle \times \langle 3 \rangle$ but I don't recognise the notation $G_1 G_2$ and I can't think what $\mathbb{Z}_6 = \langle 2 \rangle \langle 3 \rangle$ should mean.
Could someone explain that ?
Conjecture : $G_1G_2:=\{g_1G_2:g_1\in G_1\}$.
We call it the set product. It's from formal languages. It just means
$$XY=\{xy\mid x\in X, y\in Y\}.$$
We evaluate it under the operation of the group, denoted here by concatenation.
For example, in $G=\Bbb Z_2\times\Bbb Z_2=\{e,a,b,ab\}$, where $a^2=b^2=e$ and $ab=ba$, if $H=\{e,a\}$ and $K=\{e,b\}$, then
$$\begin{align} HK&=\{hk\mid h\in H, k\in K\}\\ &=\{ee,eb,ae, ab\}\\ &=\{e, b, a, ab\}\\ &=G. \end{align}$$