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).

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.

R-Algebraic Structure Isomorphism
If $$\left({S, \ast_1, \ast_2, \ldots, \ast_n: \circ}\right)_R$$ and $$\left({T, \odot_1, \odot_2, \ldots, \odot_n: \otimes}\right)_R$$ are $R$-algebraic structures, then an isomorphism:
 * $$\phi: \left({S, \ast_1, \ast_2, \ldots, \ast_n: \circ}\right)_R \to \left({T, \odot_1, \odot_2, \ldots, \odot_n: \otimes}\right)_R$$

is called an $$R$$-algebraic structure 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 $$\left({T, \preccurlyeq}\right)$$.

Isomorphic Copy
Let $$\phi: S \to T$$ be an isomorphism.

Let $$x \in S$$.

Then $$\phi \left({x}\right) \in T$$ is known as the isomorphic copy of $$x$$ (under $$\phi$$).

Group definition

 * : $$\S 7.1$$
 * : $$\S 1.5$$
 * : Chapter $$\text{II}$$
 * : $$\S 28 \gamma$$
 * : $$\S 46, \S 47 \text { (c)}$$

Ring definition

 * : $$\S 57$$ Remarks: $$\text{(a) (3), (b)}$$