Definition
A subset {aα} of an abelian group is linearly independent (over Z) if the only linear combination of these elements that is equal to zero is trivial: if
where all but finitely many coefficients nα are zero (so that the sum is, in effect, finite), then all coefficients are 0. Any two maximal linearly independent sets in A have the same cardinality, which is called the rank of A.
Rank of an abelian group is analogous to the dimension of a vector space. The main difference with the case of vector space is a presence of torsion. An element of an abelian group A is classified as torsion if its order is finite. The set of all torsion elements is a subgroup, called the torsion subgroup and denoted T(A). A group is called torsion-free if it has no non-trivial torsion elements. The factor-group A/T(A) is the unique maximal torsion-free quotient of A and its rank coincides with the rank of A.
The notion of rank with analogous properties can be defined for modules over any integral domain, the case of abelian groups corresponds to modules over Z.
Read more about this topic: Rank Of An Abelian Group
Famous quotes containing the word definition:
“... we all know the wag’s definition of a philanthropist: a man whose charity increases directly as the square of the distance.”
—George Eliot [Mary Ann (or Marian)
“The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.”
—William James (1842–1910)
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)