Isomorphism (Abstract Algebra)/Examples

Examples of Isomorphisms

$\struct {\Z \sqbrk {\sqrt 3}, +}$ with Numbers of Form $2^m 3^n$ under $\times$

Let $\Z \sqbrk {\sqrt 3}$ denote the set of quadratic integers over $3$:

$\Z \sqbrk {\sqrt 3} = \set {a + b \sqrt 3: a, b \in \Z}$

Let $S$ be the set defined as:

$S := \set {2^m 3^n: m, n \in \Z}$

Let $\struct {\Z \sqbrk {\sqrt 3}, +}$ and $\struct {S, \times}$ be the algebraic structures formed from the above with addition and multiplication respectively.

Then $\struct {\Z \sqbrk {\sqrt 3}, +}$ and $\struct {S, \times}$ are isomorphic.

$\struct {\N, +}$ under Doubling

Let $\N$ denote the set of natural numbers.

Let $2 \N$ denote the set of even non-negative integers:

$2 \N := \set {0, 2, 4, 6, \ldots}$

Let $\struct {\N, +}$ and $\struct {2 \N, +}$ be the algebraic structures formed from the above with addition.

Let $f: \N \to 2 \N$ be the mapping defined as:

$\forall n \in \N: \map f n = 2 n$

Then $f$ is an isomorphism.