Every subgroup is a direct factor iff direct product of elementary Abelian groups

Statement
The following are equivalent for a group:


 * 1) Every subgroup of the group is a direct factor.
 * 2) The group is either trivial or a direct product of elementary Abelian groups. In other words, it is a direct product of Abelian $$p$$-groups for possibly different primes $$p$$ where each $$p$$-group is an elementary Abelian group.

The implication (2) implies (1) requires the use of the axiom of choice.