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:
\(\ds e\) | \(=\) | \(\ds e \cdot e\) | where $e$ is the identity of $G$, $G_1$ and $G_2$ | |||||||||||
\(\ds x\) | \(=\) | \(\ds \paren {x^2}^2 \cdot x^3\) | ||||||||||||
\(\ds x^2\) | \(=\) | \(\ds x^2 \cdot e\) | ||||||||||||
\(\ds x^3\) | \(=\) | \(\ds e \cdot x^3\) | ||||||||||||
\(\ds x^4\) | \(=\) | \(\ds \paren {x^2}^2 \cdot e\) | ||||||||||||
\(\ds x^5\) | \(=\) | \(\ds x^2 \cdot x^3\) |
and so the conditions are fulfilled for $G_1 \times G_2$ to be an internal group direct product.
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $13$: Direct products: Direct products: Example $13.4$