# Internal Group Direct Product/Examples/C2 x C3

## Example of Internal Group Direct Product

The direct product of the cyclic groups $C_2$ and $C_3$ is isomorphic to the cyclic groups $C_6$.

Hence it is seen to be an internal group direct product.

## Proof

Let $x$ be a generator for $C_6$.

Let $G_1$ be the subgroup of $C_6$ generated by $x^2$.

Let $G_2$ be the subgroup of $C_6$ generated by $x^3$.

From Subgroup of Abelian Group is Normal, both $G_1$ and $G_2$ are normal in $G$.

Then we have:

 $\displaystyle e$ $=$ $\displaystyle e \cdot e$ where $e$ is the identity of $G$, $G_1$ and $G_2$ $\displaystyle x$ $=$ $\displaystyle \paren {x^2}^2 \cdot x^3$ $\displaystyle x^2$ $=$ $\displaystyle x^2 \cdot e$ $\displaystyle x^3$ $=$ $\displaystyle e \cdot x^3$ $\displaystyle x^4$ $=$ $\displaystyle \paren {x^2}^2 \cdot e$ $\displaystyle x^5$ $=$ $\displaystyle x^2 \cdot x^3$

and so the conditions are fulfilled for $G_1 \times G_2$ to be an internal group direct product.

$\blacksquare$