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

Let $\left({G, \circ}\right)$ and $\left({H, *}\right)$ be groups.

Let $\phi: G \to H$ be a (group) homomorphism.

Then $\phi$ is a group isomorphism iff $\phi$ is a bijection.

Ring Isomorphism

Let $\left({R, +, \circ}\right)$ and $\left({S, \oplus, *}\right)$ be rings.

Let $\phi: R \to S$ be a (ring) homomorphism.

Then $\phi$ is a ring isomorphism iff $\phi$ is a bijection.

$F$-Isomorphism

Let $R, S$ be rings with unity.

Let $F$ be a subfield of both $R$ and $S$.

Let $\varphi: R \to S$ be an $F$-homomorphism such that $\varphi$ is bijective.

Then $\varphi$ is an $F$-isomorphism.

The relationship between $R$ and $S$ is denoted $R \ \cong_F \ S$.

Field Isomorphism

Let $\left({F, +, \circ}\right)$ and $\left({K, \oplus, *}\right)$ be fields.

Let $\phi: F \to K$ be a (field) homomorphism.

Then $\phi$ is a field isomorphism iff $\phi$ is a bijection.

R-Algebraic Structure Isomorphism

Let $\left({S, \ast_1, \ast_2, \ldots, \ast_n, \circ}\right)_R$ and $\left({T, \odot_1, \odot_2, \ldots, \odot_n, \otimes}\right)_R$ be $R$-algebraic structures.

Let $\phi: S \to T$ be an $R$-algebraic structure homomorphism.

Then $\phi$ is an $R$-algebraic structure isomorphism iff $\phi$ is a bijection.

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 $\mathcal C$.

Field Isomorphism

Let $\left({F, +, \circ}\right)$ and $\left({K, \oplus, *}\right)$ be fields.

Let $\phi: F \to K$ be a (field) homomorphism.

Then $\phi$ is a field isomorphism iff $\phi$ is a bijection.

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

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.

Sources

• Iain T. Adamson: Introduction to Field Theory (1964): Chapter $1 \ \S 3$
• Seth Warner: Modern Algebra (1965): $\S 6$

Ordered Structure definition

• Seth Warner: Modern Algebra (1965): $\S 15$