Does (Z, +) have two generators but infinitely many generating sets?

Question

We say the group of integers under addition Z has only two generators, namely 1 and -1.

However, Z can also be generated by any set of 'relatively prime' integers. (Integers having gcd 1).

I have two questions here. Couldn't find a satisfactory answer anywhere.

1. If a group is generated by a set consisting of a single element, only then is it cyclic?

2. Does 'generator' mean a single generating element?

3. Is it correct to say '(Z, +) has two generators but infinitely many generating sets'?

Thank you for your help!

Answer 1

The answer to all your questions is yes. By definition a cyclic group is a group which is generated by a single element (or equivalently, by a subset containing only one element). Such an element is called a generator.

$(\mathbf{Z},+)$ of course has infinitely many generating subsets, be it only because any subset containing $1$ or $-1$ is generating, and there are of course infinitely many such subsets. There are more interesting generating subsets however, such as those containing two relatively prime integers.

Answer 2

You are correct. $(\mathbb{Z},+)$ is a cyclic group with generator $(\{+1\})$ or $(\{-1\})$ And it has many generating subgroup. May be it is important to just notice that $(\mathbb{Z},+)$ is cyclic group because it can be generated by a single element.