A finitely generated group that contains a subgroup which is not finitely generated.

Question

Question: Give a finitely generated group that contains a subgroup which is not finitely generated.

What I know: I learned that the free group with two generators $F_{2}$ (with 2 by 2 matrix generators) is finitely generated. But I do not know what the subgroup of it which is not finitely generated. Can some one help me find such a subgroup? I would appreciate!

It's not hard to show that *any* countable group can be embedded in a finitely generated group. So you can get an example like you want by taking any non-finitely-generated countable group (for example the additive group of the rational numbers) and embedding it in a finitely generated group. — bof, Jul 31 '18 at 00:01
Possible duplicate of Subgroups of finitely generated groups are not necessarily finitely generated — Lee Mosher, Jul 31 '18 at 00:10

score 5 · Answer 1 · edited Jul 30 '18 at 23:46

The free group on countable many generators, $F_{\infty}$, is isomorphic to a subgroup of $F_2$. Write $F_2 = \langle x,y\rangle$ and take $F_{\infty}$ to be the subgroup generated freely by $yxy^{-1}, y^2xy^{-2}, y^3xy^{-3} \ldots$.

There is a nice way to see this from the theory of covering spaces. Take $F_2$ to be the fundamental group of the wedge of two circles labelled $x$ and $y$. Then consider the covering space that consists of basepoints indexed by $\mathbb{Z}$, with each basepoint $n$ having two edges labelled $x$ and $y$ going to the point $n+1$.

A finitely generated group that contains a subgroup which is not finitely generated.

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