Let (G, ∘) and (G′, *) be groups, so that ∘ is the operation on G and * is the operation on G′. An isomorphism between (G, ∘) and (G′, *) is a one‐to‐one correspondence f from the set G to the set G′ such that, for all a and b in G,

This means that if f maps a to a′ and b to b′, then f maps a ∘ b to a′ * b′. So an isomorphism is a bijective homomorphism. f can be viewed as a relabelling of the Cayley table of G to become the Cayley table of G′.

If there is an isomorphism between two groups, the two groups are isomorphic to each other. This is written G≅G′. Two groups that are isomorphic to one another have essentially the same structure; the actual elements of one group may be quite different objects from the elements of the other, but the way in which they behave with respect to the operation is the same. For example, the group 1, i, −1, −i with multiplication is isomorphic to the group of elements 0, 1, 2, 3 with addition modulo 4. See appendix 17 for groups, of order 15 or less, up to isomorphism.