Finitely generated, non abelian, infinite group

Question

I was making a diagram of different types of groups; finite /infinite, cyclic / non-cyclic, finitely generated / inifinitely generated, but realized that I didn't have any examples og infinite groups, that are both finitely generated and non -abelian. Does anyone have any examples? :)

I was thinking about creating an example based on matrices and matrix multiplication, but I didn't get very far. I know that since I am looking for a finitely generated group, It must be countable.

Answer 1

The group $\langle a, b\rangle$ is

1. finitely generated: obviously, it is generated by $\{a, b\}$,
2. non-abelian: the elements $ab$ and $ba$ are two distinct elements,
3. infinite: The mapping $\mathbb N\to \langle a, b\rangle$ that maps $n$ to $a^n$ is injective.

Answer 2

Try $G=\mathbb Z \times S_3$.

Answer 3

For the curious, John Meier's Groups, Graphs and Trees as well as Office Hours With A Geometric Group Theorist edited by Matt Clay and Dan Margalit make great introductions to the world of infinite, finitely-generated groups!