# Free Abelian group

A group that is free in the variety of all Abelian groups (see Free algebra). Direct sums (of a finite or an infinite number) of infinite cyclic groups, and only these, are free in the class of Abelian groups. Here the collection of generating elements of all the cyclic direct summands is a system of free generators (also called a base) of the free Abelian group. Not every maximal linearly independent system of elements of a free Abelian group is a base for it. Free Abelian groups are isomorphic if and only if their bases have the same cardinality. The cardinality of a base of a free Abelian group coincides with the Prüfer rank of the group. Every non-null subgroup of a free Abelian group is also free. An Abelian group is free if and only if it has an ascending sequence of subgroups (see Subgroup series) each factor of which is isomorphic to an infinite cyclic group.

#### References

[1] | A.G. Kurosh, "The theory of groups" , 1–2 , Chelsea (1955–1956) (Translated from Russian) |

[2] | M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian) |

#### Comments

For Prüfer rank see Abelian group.

#### References

[a1] | L. Fuchs, "Infinite abelian groups" , 1 , Acad. Press (1970) pp. Chapt. 4 |

**How to Cite This Entry:**

Free Abelian group. O.A. Ivanova (originator),

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Free_Abelian_group&oldid=18959