<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

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

{"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

Data publikacji: 21 wrz 2017

Zakres stron: 5 - 14

Otrzymano: 31 lip 2014

Przyjęty: 20 lut 2015

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

Słowa kluczowe
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

Data publikacji: 21 wrz 2017

Zakres stron: 5 - 14

Otrzymano: 31 lip 2014

Przyjęty: 20 lut 2015

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

Słowa kluczoweGroups, Basis property, Exchange property

© 2017

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

Słowa kluczowe
Groups, Basis property, Exchange property