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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.

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