A non co-hopfian simple group?

Question

Is there a simple group (all normal subgroups are the group itself or the trivial group) that is non co-hopfian (there exists an injective group endomorphism that is not surjective) ?

Answer 1

Yes, there is. You can find a finitary alternating group $G$ (which is necessarily simple) with a proper subgroup $H$ which is isomorphic to $G$.

I found this example by googling "co-hopfian simple" and using this link to a groupprops page which was nearly at the top.

Answer 2

To answer Derek Holt's request: R. Thompson's group $V$ is a non-cohopfian finitely presented group. This is the group of right-continuous permutations of $\mathbf{R}/\mathbf{Z}$ that are, outside a finite subset of $\mathbf{Z}[1/2]/\mathbf{Z}$, affine with slopes in $\{2^n:n\in\mathbf{Z}\}$. Then the pointwise stabilizer of $[1/2,1[$ is clearly isomorphic to $V$ (conjugating by $x\mapsto 2x$ yields an isomorphism). Furthermore, it is dis-cohopfian, in the sense that it has an injective endomorphism $f$ such that $\bigcap_{n\ge 0}\mathrm{Im}(f^n)=\{1\}$.

That $V$ is simple and finitely presented is classical (historically, it's the first such infinite example).