A General Basis Theorem

Extract

The well-known “basis theorem” of elementary algebra states that in a finite-dimensional vector space, any two bases have the same number of elements; or, equivalently, that a vector space is n-dimensional if it has a basis consisting of n vectors (where the dimension of a vector space is defined to be the least upper bound of the numbers k for which there exist k linearty independent vectors, and a basis is defined to be a maximal set of linearly independent vectors). This theorem has an analogue in the theory of groups : if an Abelian group has a finite maximal set of independent non-cyclic elements, the number of elements in one such set being n, then no set of independent non-cyclic elements can have more than n members.