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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.

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    $\begingroup$ $\mathbb{Z}$ isn't it ? $\endgroup$ – ZAF Apr 9 at 11:13
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    $\begingroup$ @ZAF $\mathbb Z$ is abelian. $\endgroup$ – 5xum Apr 9 at 11:16
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    $\begingroup$ The question body and title disagree. $\endgroup$ – Randall Apr 9 at 11:46
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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.
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  • $\begingroup$ 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"? $\endgroup$ – AfterMath Apr 9 at 11:32
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    $\begingroup$ @AfterMath free group of rank 2 $\endgroup$ – Randall Apr 9 at 11:44
  • $\begingroup$ @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) $\endgroup$ – 5xum Apr 9 at 11:47
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Try $G=\mathbb Z \times S_3$.

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  • $\begingroup$ Thanks! Good example :) $\endgroup$ – AfterMath Apr 9 at 11:54
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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!

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