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).
(g1,h1) ∘ (g2,h2) = (g1 ρ g2, h1 τ h2)
These concepts can be generalized to deal with the direct product of any finite number of groups on which there are specified group operations.