In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.Definition and notationGiven two groups (G, *) and (H, \odot), a group isomorphism from (G, *) to (H, \odot) is a bijective group homomorphism from G to H. Spelled out, this means that a group isomorphism is a bijective function f : G \to H such that for all u and v in G it holds that
f(u * v) = f(u) \odot f(v).The two groups (G, *) and (H, \odot) are isomorphic if there exists an isomorphism f
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