I would like to understand the following result:
Let $G$ be a group and $A_1, A_2 \subseteq G$ be subgroups satisfying the following conditions
(a) For all $g \in G$ there are elements $h_1 \in A_1$ and $h_2 \in A_2$ such that $g=h_1 h_2$.
(b) $A_1 \cap A_2 = \{ e \}$
(c) For all $h_1 \in A_1$ and $h_2 \in A_2$ we have $h_1 h_2 = h_2 h_1$.
Then $G$ is isomorphic to $A_1 \times A_2$.
In order to understand this result, I am trying to find some examples of subgroups satisfying this property.
My examples:
- If $G = A_1$ and $A_2 = \{ e \}$, the properties are obviously satisfied.
- If $V$ is a vectorspace and $G = (V,+)$ and $A_1,A_2$ are two subspaces of $V$ satisfying $V = A_1 \oplus A_2$ (in the context of direct sums of vector spaces), then these properties are also satisfied.
However, I am not happy with my found examples yet. I would like have an example where
- $G$ is a non-abelian group and
- where neither $A_1$ nor $A_2$ is the trivial group.
Could you please help me finding such an example? Thank you very much!