A characterization of alternating groups

It is proved that a group $G$ generated by a conjugacy class $X$ of elements of order 3, so that every two non-commuting elements of $X$ generate a subgroup isomorphic to an alternating group of degree 4 or 5, is locally finite. More precisely, either $G$ contains a normal elementary 2-subgroup of index 3, or $G$ is isomorphic to an alternating group of permutations on some (possibly infinite) set.