# Easy example of an infinite simple group with an embedding into a finitely presented group

I would like an easy example of an infinite simple group along with an embedding into a finitely presented group. I know that there are infinite simple finitely presented group such as Thompson’s group $$V$$ and I know that any group with solvable word problem embeds into a simple subgroup of a finitely presented group, but I am looking for an example of a more elementary nature.

• One example is Burger-Mozes's group in "Finitely presented simple groups and products of trees", ethz.ch/content/dam/ethz/special-interest/math/department/…. Aug 13 '19 at 0:22
• And there are many more examples, see en.wikipedia.org/wiki/Simple_group#Infinite_simple_groups. Aug 13 '19 at 0:29
• That Burger-Mozes groups are finitely presented is somewhat immediate once constructed. But the construction, and the proof of simplicity are both subtle.
– YCor
Aug 13 '19 at 6:07
• The group $A$ of alternating finitely supported permutations of an infinite countable set embeds into Houghton's group $H_3$. The latter is quite easy to construct, and that it is finitely generated and contains $A$ is easy, and that $A$ is simple is also easy. That $H_3$ is finitely presented is more technical with some combinatorial arguments.
– YCor
Aug 13 '19 at 6:09
• So you're looking for a simple simple group? Aug 13 '19 at 6:57

In particular, Section 3 contains a short and elementary proof of the fact that the group $$S_\infty$$ of bijections of $$\mathbb{N}$$ with finite supports (which contains the infinite simple group $$A_\infty$$ of alternating bijections) embeds into a finitely presented group. A presentation for such a group, simplified in Section 4, is $$\langle a,b,x \mid a^2=1, (xaxa^{-1})^3=1, [x,a^2xa^{-2}]=1, x=[a,b], axa^{-1}=bxb^{-1} \rangle.$$ Actually, the group defined by the presentation coincides with Houghton's group $$H_3$$ already mentioned by Yves in the comments.
• To have a complete proof of an embedding, one needs to check that $[x,a^nxa^{-n}]=1$ in this group for all $n\ge 2$.