<abstract xmlns="http://www.w3.org/1999/xhtml"><p> 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. </p></abstract>

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.

Relation Between Groups with Basis Property and Groups with Exchange Property

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

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{"article-title":"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....

Relation Between Groups with Basis Property and Groups with Exchange Property

Published Online: Sep 21, 2017

Page range: 5 - 14

Received: Jul 31, 2014

Accepted: Feb 20, 2015

DOI: https://doi.org/10.1515/auom-2016-0024

Keywords
Groups, Basis property, Exchange property

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Relation Between Groups with Basis Property and Groups with Exchange Property

Published Online: Sep 21, 2017

Page range: 5 - 14

Received: Jul 31, 2014

Accepted: Feb 20, 2015

DOI: https://doi.org/10.1515/auom-2016-0024

KeywordsGroups, Basis property, Exchange property

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Keywords
Groups, Basis property, Exchange property