Compatible Group Operations.

Let X be a nonempty set, and suppose that ⋅ and ⊙ are binary operations on X such that, for all a, to, c ∈ X, (a ⋅ b) ⊙ c = a ⋅ (b ⊙ c) and (a ⊙ c) ⋅ c = a ⊙ (b ⋅ c). Then we say that ⋅ and ⊙ are compatible. In this article, we show that such operations arise on disjoint unions of isomorphic groups. In fact, given disjoint groups G and H, we show that G is isomorphic to H if and only if the group operations of G and H extend, respectively, to compatible group operations ⋅, ⊙ on G ∪ H. [ABSTRACT FROM AUTHOR]