Group Isomorphism/Examples

Examples of Group Isomorphism

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 $\alpha \ne 1$.

Congruence Modulo Initial Segment of Natural Numbers

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Let $\N_{<m}$ denote the initial segment of the natural numbers $\N$:

$\N_{<m} = \set {0, 1, \ldots, m - 1}$

Let $\RR_m$ denote the equivalence relation:

$\forall x, y \in \Z: x \mathrel {\RR_m} y \iff \exists k \in \Z: x = y + k m$

For each $a \in \N_{<m}$, let $\eqclass a m$ be the equivalence class of $a \in \N_{<m}$ under $\RR_m$:

$\eqclass a m := \set {a + z m: z \in \Z}$

Let $\Z_m$ be the set defined as:

$\Z_m := \set {\eqclass a m: a \in \N_{<m} }$

Let $+_\PP$ denote the operation induced on $\powerset \Z$ by integer addition.

Let $\phi_m: \struct {\N_m, +_m} \to \struct {\Z_m, +_\PP}$ be the mapping defined by:

$\forall a \in \N_m: \map {\phi_m} a = \eqclass a m$

where $\struct {\N_m, +_m}$ denotes the additive group of integers modulo $m$.

Then $\phi_m$ is a (group) isomorphism.