Relation between Groups with Basis Property and Groups with Exchange Property

A group G is called a group with basis property if there exists a basis (minimal generating set) for every subgroup H of G and every two bases are equivalent. A group G is called a group with exchange property, if x / ∈ 〈X〉 ∧ x ∈ 〈X ∪ {y}〉 , then y ∈ 〈X ∪ {x}〉, for all x, y ∈ G and for every subset X ⊆ G. In this research, we proved the following: Every polycyclic group satisfies the basis property. Every element in a group with the exchange property has a prime order. Every p-group satisfies the exchange property if and only if it is an elementary abelian p-group. Finally, we found necessary and sufficient condition for every group to satisfy the exchange property, based on a group with the basis property.