Definition:Isomorphism (Abstract Algebra)/Group Isomorphism

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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

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