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I was wondering this today, and my algebra professor didn't know the answer.

Are subgroups of finitely generated groups also finitely generated?

I suppose it is necessarily true for finitely generated abelian groups, but is it true in general?

And if not, is there a simple example of a finitely generated group with a non-finitely generated subgroup?

NOTE: This question has been merged with another question, asked by an undergraduate. For an example not involving free groups, please see Andreas Caranti's answer, which was the accepted answer on the merged question.

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    $\begingroup$ Like you noted, this is indeed true for finitely generated abelian groups. Put another way, this is the same as the statement that every finitely generated Z-module is Noetherian, which holds since Z is Noetherian. $\endgroup$ – Jason Juett Jul 9 '16 at 11:33
  • $\begingroup$ Since there are Noetherian groups, you can guess the answer is no. $\endgroup$ – Qian Jul 31 '16 at 15:54
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It is well-known that the free group $F_2$ on two generators has as a subgroup a group isomorphic to a free group on a countably infinite set of generators. See Qiaochu's example.

However a finite index subgroup of a finitely generated group is finitely generated.

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No. The example given on Wikipedia is that the free group $F_2$ contains a subgroup generated by $y^n x y^{-n}, n \ge 1$, which is free on countably many generators.

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  • $\begingroup$ But wouldn't that subgroup still be generated by $x,y$? $\endgroup$ – Guacho Perez Jun 29 '17 at 15:56
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    $\begingroup$ @Guacho: no. $y$ is not an element of the subgroup. $\endgroup$ – Qiaochu Yuan Jun 29 '17 at 23:32
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A theorem of Higman, Neumann, and Neumann says that every countable group (no matter what horrible properties it might have) can be embedded as a subgroup of a group generated by $2$ elements. Thus subgroups of finitely generated groups can be pretty much anything.

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One of the easiest (counter)example is in Hungerford's Algebra.

Let $G$ be the multiplicative group generated by the real matrices $$a = \left(\begin{array}{l l} 1 & 1\\ 0 & 1 \end{array}\right), b = \left(\begin{array}{l l} 2 & 0\\ 0 & 1 \end{array}\right) $$ Let $H$ be the subgroup of $G$ consisting of matrices that have $1$s on the main diagonal. Then $H$ is not finitely generated.

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    $\begingroup$ isn't H just generated by a? $\endgroup$ – David Warren Katz Jul 12 '16 at 19:20
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    $\begingroup$ @DavidWarrenKatz The powers of $a$ give only integers on the $(1,2)$ entry, while $H$ has on that entry rational numbers of the form $p/2^q$. $\endgroup$ – user26857 Jun 2 at 21:33
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Perhaps an elementary example can be provided by the wreath product of two copies of (the additive group of) $\mathbf{Z}$.

Take copies $G_{i}$ of $\mathbf{Z}$, for $i \in \mathbf{Z}$, and let $$ B = \coprod_{i \in \mathbf{Z}} G_{i} $$ be the direct sum (coproduct in the category of abelian groups).

Now let another copy $H = \langle h \rangle$ of $\mathbf{Z}$ act on $B$ by $$ G_{i}^{h} = G_{i+1}. $$ More precisely, conjugation by $h$ takes a generator $g_{i}$ in the copy $G_{i}$ of $\mathbf{Z}$ to a generator $g_{i+1}$ of the $(i+1)$-th copy.

Then the semidirect product $G = B \rtimes H$ is generated by $g_{0}$ and $h$, but its subgroup $B$ requires an infinite number of generators.

It is easy to see what is going on. $B$ requires an infinite number of generators $g_{i}$. Now $h$ takes one of these generators by conjugation to all others.

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    $\begingroup$ OK, turns out that this one works! I wouldn't have thought someone would know wreath products before free groups, but I guess it's understandable they might. $\endgroup$ – Tara B Feb 19 '13 at 17:15
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    $\begingroup$ @TaraB, thanks! I was exposed to this very construction (in the elementary terms I have tried to use above, and without any mention of the term wreath product) before I learned about free groups. The key concept is that of a semidirect product. $\endgroup$ – Andreas Caranti Feb 19 '13 at 17:21
  • $\begingroup$ Yes. I don't think I even encountered semidirect products until after free groups, myself, though. In fact I know I didn't, because I definitely knew about presentations by then. $\endgroup$ – Tara B Feb 19 '13 at 17:23
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    $\begingroup$ Note that this answer and these comments were made in a different context (this question was merged with another question asked by an undergraduate who hadn't encountered free groups). $\endgroup$ – Tara B Feb 19 '13 at 18:14
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    $\begingroup$ @exfret, thanks, I have added a line to clarify that I meant the coproduct in the category of abelian groups. $\endgroup$ – Andreas Caranti Feb 15 at 13:11

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