# Finitely generated, non abelian, infinite group

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.

• $\mathbb{Z}$ isn't it ? – ZAF Apr 9 '19 at 11:13
• @ZAF $\mathbb Z$ is abelian. – 5xum Apr 9 '19 at 11:16
• The question body and title disagree. – Randall Apr 9 '19 at 11:46

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.
• Thanx alot! Guess I couldn't see the forest for the trees... May I also ask if the group <a,b> of "ordered sequences of powers of the elements a and b" has any other name than just "the group generated by the non - commutative elements a and b"? – AfterMath Apr 9 '19 at 11:32
• @AfterMath free group of rank 2 – Randall Apr 9 '19 at 11:44
• @AfterMath It's the free grop of $\{a,b\}$, or the free group of rank $2$ (since, up to isomorphism, all free groups over a set of the same cardinality are equal) – 5xum Apr 9 '19 at 11:47

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

• Thanks! Good example :) – AfterMath Apr 9 '19 at 11:54

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!