TITLE: Easy example of an infinite simple group with an embedding into a finitely presented group
QUESTION [6 upvotes]: 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.

REPLY [7 votes]: I think D. L. Johnson's article Embedding some recursively presented groups should answer your question. The abstract is:

We seek to illustrate the Higman Embedding Theorem by finding actual embeddings
  of various popular recursively presented groups in finitely presented ones,
  and are successful in at least one case.

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.