Integers under Addition form Infinite Cyclic Group

Theorem

The additive group of integers $\struct {\Z, +}$ is an infinite cyclic group which is generated by the element $1 \in \Z$.

Proof

By Epimorphism from Integers to Cyclic Group and integer multiplication:

$\forall n \in \Z: n = \map {+^n} 1 \in \gen 1$

Thus:

$\struct {\Z, +} = \gen 1$

and thus, by the definition of a cyclic group, is cyclic.

$\blacksquare$

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 5.3$. Subgroup generated by a subset: Example $98$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $25$. Cyclic Groups and Lagrange's Theorem: Theorem $25.2$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{II}$: Groups: A Little Number Theory
• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 43$
• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\text{I}$: Groups: $\S 1$: Semigroups, Monoids and Groups
• 1974: Thomas W. Hungerford: Algebra ... (previous) ... (next): $\S 1.2$
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 3.4$: Cyclic groups