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direct product

direct product (product group) of two groups G and H with group operations ρ and τ respectively. The group consisting of the elements in the Cartesian product of G and H and on which there is a dyadic operation ◦ defined as follows: (g1,h1) ◦ (g2,h2) = (g1 ρ g2, h1 τ h2)

The identity of this group is then just (eG,eH), where eG and eH are the identities of groups G and H respectively. The inverse of (g,h) is then (g–1,h–1).

These concepts can be generalized to deal with the direct product of any finite number of groups on which there are specified group operations.

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