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Revision difference : external direct product of groups |
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The \emph{external direct product} $G \times H$ of two groups $G$ and $H$ is defined to be the set of ordered pairs $(g,h)$, with $g\in G$ and $h\in H$. The group operation is defined by
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The external direct product $G \times H$ of two groups $G$ and $H$ is defined to be the set of ordered pairs $(g,h)$, with $g\in G$ and $h\in H$. The group operation is defined by
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| $(g,h)(g',h') = (gg', hh')$ |
$(g,h)(g',h') = (gg', hh')$ |
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| It can be shown that $G \times H$ obeys the group axioms. More generally, we can define the external direct product of $n$ groups, in the obvious way. Let $G = G_1 \times \ldots \times G_n$ be the set of all ordered n-tuples $\{(g_1, g_2 \ldots ,g_n) \mid g_i \in G_i\}$ and define the group operation by componentwise multiplication as before. |
It can be shown that $G \times H$ obeys the group axioms. More generally, we can define the external direct product of $n$ groups, in the obvious way. Let $G = G_1 \times \ldots \times G_n$ be the set of all ordered n-tuples $\{(g_1, g_2 \ldots ,g_n) \mid g_i \in G_i\}$ and define the group operation by componentwise multiplication as before. |
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