Definition:Isomorphism (Abstract Algebra)

Definition
An isomorphism is a homomorphism which is a bijection.

That is, it is a mapping which is both a monomorphism and an epimorphism.

An algebraic structure $\left({S, \circ}\right)$ is isomorphic to another algebraic structure $\left({T, *}\right)$ iff there exists an isomorphism from $\left({S, \circ}\right)$ to $\left({T, *}\right)$, and we can write $S \cong T$ (although notation may vary).

Also see

 * Automorphism (Abstract Algebra)

Linguistic Note
The word isomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.

Thus isomorphism means equal structure.