Group isomorphism

Given two mathematical groups (G, *) and (H, @) a group isomorphism from (G, *) to (H, @) is an isomorphism that preserves the operation, that is, it is a bijection h : G -> H such that for all u and v in G it holds that

h(u) @ h(v) = h(u * v).

If there is a group isomorphism between two groups then these groups are called group isomorphic or simply isomorphic.