Epimorphism from Integers to Cyclic Group

Theorem

Let $\gen a = \struct {G, \circ}$ be a cyclic group.

Let $f: \Z \to G$ be a mapping defined as:

$\forall n \in \Z: \map f n = a^n$.

Then $f$ is a (group) epimorphism from $\struct {\Z, +}$ onto $\gen a$.

Proof

By Powers of Element form Subgroup:

$\forall n \in \N: a^n \in \gen a$

Hence by the Index Law for Monoids: Negative Index:

$\forall n \in \Z: a^n \in \gen a$

Also, by Index Law for Monoids: Sum of Indices, $f$ is a homomorphism from $\struct {\Z, +}$ into $\struct {G, \circ}$.

By Homomorphism Preserves Subsemigroups, its codomain $f \sqbrk \Z$ is therefore a subgroup of $\gen a$ containing $a$.

By Existence of Unique Subgroup Generated by Subset: Singleton Generator, $\gen a$ is the smallest subgroup of $G$ containing $a$.

So:

$f \sqbrk \Z = \gen a$

Therefore $f$ is an epimorphism from $\struct {\Z, +}$ onto $\struct {G, \circ}$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem: Theorem $25.1$