Definition:Group Direct Product
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Definition
Let $\left({G, \circ_1}\right)$ and $\left({H, \circ_2}\right)$ be groups.
The (external) direct product of $\left({G, \circ_1}\right)$ and $\left({H, \circ_2}\right)$ is the set of ordered pairs:
- $\left({G \times H, \circ}\right) = \left\{{\left({g, h}\right): g \in G, h \in H}\right\}$
where the operation $\circ$ is defined as:
- $\left({g_1, h_1}\right) \circ \left({g_2, h_2}\right) = \left({g_1 \circ_1 g_2, h_1 \circ_2 h_2}\right)$
This is usually referred to as the group direct product of $G$ and $H$.
Generalized Definition
Let $\left({G_1, \circ_1}\right), \left({G_2, \circ_2}\right), \ldots, \left({G_n, \circ_n}\right)$ be groups.
Let $\displaystyle G = \prod_{k=1}^n G_k$ be as defined in generalized cartesian product.
The operation induced on $G$ by $\circ_1, \ldots, \circ_n$ is the operation $\circ$ defined by:
- $\left({g_1, g_2, \ldots, g_n}\right) \circ \left({h_1, h_2, \ldots, h_n}\right) = \left({g_1 \circ_1 h_1, g_2 \circ_2 h_2, \ldots, g_n \circ_n h_n}\right)$
for all ordered $n$-tuples in $G$.
The group $\left({G, \circ}\right)$ is called the (external) direct product of $\left({G_1, \circ_1}\right), \left({G_2, \circ_2}\right), \ldots, \left({G_n, \circ_n}\right)$.
Comment
Although this is just a more specific example of the external direct product of general algebraic structures, it is usually defined and treated separately because of its considerable conceptual importance.
Note that $G$ and $H$ etc. are not subsets of $G \times H$ and therefore are not subgroups of it either.
Also see
- Results about group direct products can be found here.
Sources
- George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Chapter $\text{II}$
- Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 26 \zeta, \ 26 \nu$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): Exercise $6.1$
- John F. Humphreys: A Course in Group Theory (1996): $\S 1$: Example $1.10$
