PlanetMath: external direct product of groups

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external direct product of groups

(Definition)

The external direct product $G \times H$ of two groups and is defined to be the set of ordered pairs , with $g\in G$ and $h\in H$ . The group operation is defined by

It can be shown that $G \times H$ obeys the group axioms. More generally, we can define the external direct product of 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.

"external direct product of groups" is owned by mathcam. [ full author list (2) | owner history (1) ]

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Other names:

direct product

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Cross-references: multiplication, n-tuples, obvious, axioms, group operation, ordered pairs, groups
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This is version 5 of external direct product of groups, born on 2002-02-19, modified 2005-08-26.
Object id is 2180, canonical name is DirectProduct2.
Accessed 6658 times total.

Classification:

AMS MSC:

20K25 (Group theory and generalizations :: Abelian groups :: Direct sums, direct products, etc.)

Pending Errata and Addenda

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