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.

Field Isomorphism
If both $\left({F_1, +, \circ}\right)$ and $\left({F_2, \oplus, *}\right)$ are fields, then an isomorphism:
 * $\phi: \left({F_1, +, \circ}\right) \to \left({F_2, \oplus, *}\right)$

is called a field 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.

Note that this definition also applies to modules and also to vector spaces.

Category Theory
Let $\mathcal C$ be a category, and let $X,Y$ be objects of $\mathcal C$.

A morphism $f : X \to Y$ is an isomorphism if there exists a morphism $g : Y \to X$ such that


 * $g \circ f = \operatorname{id}_X$, and $f \circ g = \operatorname{id}_Y$

where $\operatorname{id}_Z$ denotes the identity morphism on an object $Z$ of $\mathcal C$.

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 $\phi$).

Also see

 * Automorphism

Group definition

 * : $\S 7.1$
 * : $\S 1.5$
 * : Chapter $\text{II}$
 * : $\S 28 \gamma$
 * : $\S 46, \ \S 47 \ \text{(c)}$
 * : $\S 2$: Example $2.19$
 * : $\S 8$: Definition $8.8$

Ring definition

 * : $\S 23$
 * : $\S 5.24$
 * : $\S 2.2$: Definition $2.4$
 * : $\S 57$ Remarks: $\text{(a) (3), (b)}$

R-Algebraic Structure definition

 * : $\S 26$

Ordered Structure definition

 * : $\S 15$