Definition:Isomorphism (Abstract Algebra)/Group Isomorphism
Definition
Let $\struct {G, \circ}$ and $\struct {H, *}$ be groups.
Let $\phi: G \to H$ be a (group) homomorphism.
Then $\phi$ is a group isomorphism if and only if $\phi$ is a bijection.
That is, $\phi$ is a group isomorphism if and only if $\phi$ is both a monomorphism and an epimorphism.
If $G$ is isomorphic to $H$, then the notation $G \cong H$ can be used (although notation varies).
Also known as
Isomorphism as defined here is known by some authors as simple isomorphism.
Examples
Order $2$ Matrices with $1$ Real Variable
Let $S$ be the set defined as:
- $S := \set {\begin{bmatrix} 1 & t \\ 0 & 1 \end{bmatrix}: t \in \R}$
Consider the algebraic structure $\struct {S, \times}$, where $\times$ is used to denote (conventional) matrix multiplication.
Then $\struct {S, \times}$ is isomorphic to the additive group of real numbers $\struct {\R, +}$.
$\Z / 3 \Z$ With $A_4 / K_4$
Let $\Z / 3 \Z$ denote the quotient group of the additive group of integers by the additive group of $3 \times$ the integers.
Let $A_4 / K_4$ denote the quotient group of the alternating group on 4 letters by the Klein $4$-group.
Then $\Z / 3 \Z$ is isomorphic to $A_4 / K_4$.
Real Power Function
Let $\struct {\R, +}$ be the additive group of real numbers.
Let $\struct {\R_{>0}, \times}$ be the multiplicative group of positive real numbers.
Let $\alpha \in \R_{>0}$ be a strictly positive real number.
Let $f: \struct {\R, +} \to \struct {\R_{> 0}, \times}$ be the mapping:
- $\forall x \in \R: \map f x = \alpha^x$
where $\alpha^x$ denotes $\alpha$ to the power of $x$.
Then $f$ is a (group) automorphism if and only if $\alpha \ne 1$.
Also see
- Results about group isomorphisms can be found here.
Linguistic Note
The word isomorphism derives from the Greek morphe (μορφή) meaning form or structure, with the prefix iso- meaning equal.
Thus isomorphism means equal structure.
Sources
- 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 7$: Isomorphic Groups
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 7.1$. Homomorphisms
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 1.5$
- 1967: John D. Dixon: Problems in Group Theory ... (previous) ... (next): Introduction: Notation
- 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: Quotient Groups
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Introduction
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 28 \gamma$
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Group Homomorphism and Isomorphism: $\S 62$
- 1978: John S. Rose: A Course on Group Theory ... (previous) ... (next): $0$: Some Conventions and some Basic Facts
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 46$. Isomorphic groups
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 47.5 \ \text{(c)}$ Homomorphisms and their elementary properties
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.2$: Groups; the axioms
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $8$: The Homomorphism Theorem: Definition $8.8$
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): isomorphism (of groups)