List of Elements in Infinite Cyclic Group

Theorem

Let $\gen a$ be the infinite cyclic subgroup generated by $a$.

Then:

$\set {\ldots, a^{-2}, a^{-1}, a^0, a^1, a^2, \ldots}$

is a complete repetition-free list of the elements of $\gen a$.

That is:

$G = \set {a^n: n \in \Z}$

Proof

From Infinite Cyclic Group is Isomorphic to Integers:

$\gen a \cong \struct {\Z, +}$

That is, elements of $\gen a$ correspond one-to-one with $\struct {\Z, +}$ by the bijection $\phi: \Z \to \gen a$ defined as:

$\forall x \in \Z: \map \phi x = a^x$

Hence the result.

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 43$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): generator: 2.
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): generator: 2.