Free Abelian Group

• (Fraleigh 38.1) Let be a subset of a nonzero Abelian Group . The following are equivalent ¹
  • Each nonzero element can be expressed uniquely, up to order of summands, in the form
    For in and distinct .
  • generates and for and distinct if and only if .

• An Abelian group having a generating set satisfying the conditions in (Fraleigh 38.1) is called a Free Abelian Group. is called the basis for the group.

• Another way to define it is using free groups and factor groups. Let be the basis, a free abelian group, a free group on and the commutator subgroup of . Then

• (Fraleigh 38.5) If is a nonzero free abelian group with a basis of elements, then

• (Fraleigh 38.6) Let be a free abelian group with a finite basis. Then every basis of is finite and all bases of have the same number of elements (see Fraleigh 30.20).

• If is a free abelian group, then the rank of is the number of elements in a basis for .

• (Fraleigh 38.8) Let be a finitely generated abelian group with generating set . Let

  be defined by

  Then is a homomorphism.

• (Fraleigh 38.9) Basis Replacement Theorem If is a basis for a free abelian group and . Then for , the set

  Is also a basis for . Note we removed from and replaced it with .

• (Fraleigh 38.11) Let be a nonzero abelian group of finite rank and let be a nonzero subgroup of . Then is free abelian of rank .

  Also, there exists a basis for and positive integers where divides for such that

  is a basis for .

  • Idea: Clearly a basis for is also a basis for . The general idea will be to construct a new basis in such that for all non-zero , we have

    That is, the coefficients for each is .
    The basis can be constructed iteratively. In particular, starting with a basis and a potential reordering of elements in the basis, derive, using an element a constant such that (via the division algorithm).

    Now each element of can be expressed as a sum of a multiple of and the rest of the elements in the bases. From these elements, repeat the process above and obtain . The process terminates when the only element in that remains is . Also from the repeated subtraction we have divides .

• (Fraleigh e38.10) A free abelian group contains no nonzero elements of finite order

  • Idea: Any such element will satisfy . Also, clearly . Therefore . Thus, the abelian group is not free.

Links §

• A First Course in Abstract Algebra 7th Edition by Fraleigh

Footnotes §

1. For an analogous concept, see Matroid Theory. We may think of this theorem as talking about linear independence. ↩