Definition:Isomorphism (Abstract Algebra)

An isomorphism is a homomorphism which is a bijection.

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).

Group Isomorphism
If both $$\left({G, \circ}\right)$$ and $$\left({H, *}\right)$$ are groups, then an isomorphism $$\phi: \left({G, \circ}\right) \to \left({H, *}\right)$$ is called a group isomorphism.

Ring Isomorphism
If both $$\left({R, +, \circ}\right)$$ and $$\left({S, \oplus, *}\right)$$ are rings, then an isomorphism $$\phi: \left({R, +, \circ}\right) \to \left({S, \oplus, *}\right)$$ is called a ring isomorphism.

Isomorphism on an Ordered Structure
An isomorphism from an ordered structure $$\left({S, \circ; \preceq}\right)$$ to another $$\left({T, *; \preccurlyeq}\right)$$ is a mapping $$\phi: S \to T$$ that is both:


 * An isomorphism, i.e. a bijective homomorphism, from the structure $$\left({S, \circ}\right)$$ to the structure $$\left({T, *}\right)$$;
 * An order isomorphism from the poset $$\left({S; \preceq}\right)$$ to the poset $$\left({T; \preccurlyeq}\right)$$.