Set of Integers under Addition is Isomorphic to Set of Even Integers under Addition

Theorem

Let $\struct {\Z, +}$ be the algebraic structure formed by the set of integers under the operation of addition.

Let $\struct {2 \Z, +}$ be the algebraic structure formed by the set of even integers under the operation of addition.

Then $\struct {\Z, +}$ and $\struct {2 \Z, +}$ are isomorphic.

Proof

Let $f: \Z \to 2 \Z$ be the mapping:

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

From Bijection between Integers and Even Integers, $f$ is a bijection.

Let $m, n \in \Z$.

Then:

\(\ds \map f {m + n}\)

\(=\)

\(\ds 2 \paren {m + n}\)

Definition of $f$

\(\ds \)

\(=\)

\(\ds 2 m + 2 n\)

Integer Multiplication Distributes over Addition

\(\ds \)

\(=\)

\(\ds \map f m + \map f n\)

Integer Multiplication Distributes over Addition

Thus $f$ is an isomorphism by definition.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 6$: Isomorphisms of Algebraic Structures: Example $6.4$