Internal Group Direct Product/Examples/C2 x C3

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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.

$\blacksquare$


Sources