Assertion: free abelian group | /r/HasProperty | f \displaystyle f
Facets:
Source sentences:
text: A free abelian group F {\displaystyle F} with basis B {\displaystyle B} has the following universal property: for every function f {\displaystyle f} from B {\displaystyle B} to an abelian group A {\displaystyle A} , there exists a unique group homomorphism from F {\displaystyle F} to A {\displaystyle A} which extends f {\displaystyle f} .
text: If G {\displaystyle G} is a subgroup of a finitely generated free abelian group F {\displaystyle F} , then G {\displaystyle G} is free and there exists a basis ( e 1 , … , e n ) {\displaystyle (e_{1},\ldots ,e_{n})} of F {\displaystyle F} and positive integers d
text: Given an arbitrary abelian group A {\displaystyle A} , there always exists a free abelian group F {\displaystyle F} and a surjective group homomorphism from F {\displaystyle F} to A {\displaystyle A} .