Definition:Infinite Cyclic Group/Also known as

Infinite Cyclic Group: Also known as

An infinite cyclic group is also known as a free group on one generator.

From Integers under Addition form Infinite Cyclic Group, the additive group of integers $\struct {\Z, +}$ forms an infinite cyclic group.

Thus the notation $\Z$ is often used for the infinite cyclic group.

This is justified as, from Cyclic Groups of Same Order are Isomorphic, $\Z$ is isomorphic to $\gen a$.