# Is there a residually finite group not finitely presented?

I am looking for a residually finite group which is not finitely presented. Does such a group exist?

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$G =\displaystyle\prod_{i\in I} \mathbb{Z}_2$ is not finitely generated (hence not finitely presented) whenever $I$ is infinite. Using projection homomorphisms you can show $G$ to be residually finite.

I can't think of an easy answer for finitely generated groups, but a look at the result in this paper towards the bottom of page 4 shows that that the Lamplighter Group should work.

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Thank you for your answer. A more interesting example would be a finitely generated group. –  Seirios Jul 10 '12 at 8:00
Looks like you got there before me. –  user17794 Jul 10 '12 at 8:42

Finally, I found a nice example: the lamplighter group $L_2= \mathbb{Z}_2 \wr \mathbb{Z}$. There is a natural morphism from $L_2$ to $\mathbb{Z}_2 \wr \mathbb{Z}_n$ and a such morphism, for $n$ big enough, send a non-trivial element on a non-trivial element in the finite group $\mathbb{Z}_2 \wr \mathbb{Z}_n$.

Moreover, one can show that $F \wr \mathbb{Z}$ is residually finite iff $F$ is abelian.

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