Simple not implies co-Hopfian

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., simple group) need not satisfy the second group property (i.e., co-Hopfian group)
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Statement

A simple group need not be a co-Hopfian group: it is possible to have a simple group that is isomorphic to a proper subgroup of itself.

Facts used

1. Finitary alternating groups are simple

Proof

Example of the finitary alternating group

Let $A$ be an infinite set and $B$ a proper subset of $A$ of the same cardinality. Let $\phi :A\to B$ be a bijection. Let $G$ be the finitary alternating group on $A$ and $H$ be the subgroup of $G$ comprising those permutations that fix the complement of $B$. Note that $H$ can be naturally identified with the finitary alternating group on $B$.

Then, $\phi$ induces an isomorphism from $G$ to $H$. Thus, $G$ is isomorphic to the proper subgroup $H$. by fact (1), we thus have a simple group that is isomorphic to a proper subgroup of itself.