Periodic group

Definition
A group is termed a periodic group or torsion group if it satisfies the following equivalent conditions:


 * 1) Every element of the group has finite order.
 * 2) The group is a union of finite subgroups, i.e., it is the union of a collection of subgroups, each of which is finite.
 * 3) Every submonoid of the group (i.e., every subset that contains the identity element and is closed under multiplication, making it a monoid) is a subgroup.
 * 4) Every nonempty subsemigroup of the group (i.e., every subset that is closed under multiplication) is a subgroup.

Note that we do not assume a uniform bound on the orders of all elements. Thus, the exponent of a periodic group may be finite or infinite.