Given two groups $G_1$ and $G_2$ with operations $*_1$ and $*_2$, the **direct product** of sets $G_1\times G_2$ is a group under the operation
\[ (a_1, a_2)*(b_1, b_2) = (a_1*_1 b_1, a_2 *_2 b_2). \]
This has normal subgroups
\[ H_1=\{(a,e_2) \mid a \in G_1\}\cong G_1\]
and
\[ H_2=\{(e_1,a) \mid a \in G_2\}\cong G_2\]
such that $H_1\cap H_2 = \{(e_1,e_2)\}$ and $H_1H_2=G_1\times G_2$.

Conversely, if a group $G$ has normal subgroups $H$ and $K$ such that $H\cap K=\{e\}$ and $HK=G$, then $G\cong H\times K$. We call $H$ and $K$ a **direct factors** of $G$ in this case.

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- Review status: reviewed
- Last edited by David Roe on 2021-09-30 16:48:21

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**History:**(expand/hide all)

- 2021-09-30 16:48:21 by David Roe (Reviewed)
- 2021-09-30 16:48:05 by David Roe
- 2019-05-23 18:50:36 by John Jones
- 2019-05-23 18:45:54 by John Jones

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