Group generated by finitely many periodic elements

Definition
A group is termed a group generated by finitely many periodic elements if it satisfies the following equivalent conditions:


 * 1) It has a generating set that is finite and comprises elements only of finite orders.
 * 2) It is both a finitely generated group and a group generated by periodic elements (this definition is a priori different from the first one in that we are not assuming that the finite generating set and the generating set comprising periodic elements are the same).
 * 3) It is a join of finitely many finite subgroups.
 * 4) It is a join of finitely many finite cyclic subgroups.