Assertion: abelian group's subgroup | /r/HasProperty | free abelian
Facets:
value: (below)
type: OTHER
Source sentences:
text: Every subgroup of a free Abelian group is free Abelian.
text: This result of Richard Dedekind was a precursor to the analogous Nielsen–Schreier theorem that every subgroup of a free group is free, and is a generalization of the fact that every nontrivial subgroup of the infinite cyclic group is infinite cyclic.
text: Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups.
text: This fact, together with the fact that every subgroup of a free abelian group is free abelian (below) can be used to show that every finitely generated abelian group is finitely presented.
text: But since this subgroup is itself free abelian, it is also finitely generated, and its basis (together with the commutators over B) forms a finite set of relators for a presentation of G.
text: The Nielsen–Schreier theorem: Every subgroup of a free group is free.
text: For example, one can show that every subgroup of a free group is free.