Rank of An Abelian Group - Groups of Higher Rank

Groups of Higher Rank

Abelian groups of rank greater than 1 are sources of interesting examples. For instance, for every cardinal d there exist torsion-free abelian groups of rank d that are indecomposable, i.e. cannot be expressed as a direct sum of a pair of their proper subgroups. These examples demonstrate that torsion-free abelian group of rank greater than 1 cannot be simply built by direct sums from torsion-free abelian groups of rank 1, whose theory is well-understood. Moreover, for every integer n ≥ 3, there is a torsion-free abelian group of rank 2n − 2 that is simultaneously a sum of two indecomposable groups, and a sum of n indecomposable groups. Hence even the number of indecomposable summands of a group of an even rank greater or equal than 4 is not well-defined.

Another result about non-uniqueness of direct sum decompositions is due to A.L.S. Corner: given integers n ≥ k ≥ 1, there exists a torsion-free abelian group A of rank n such that for any partition n = r₁ + ... + r_k into k natural summands, the group A is the direct sum of k indecomposable subgroups of ranks r₁, r₂, ..., r_k. Thus the sequence of ranks of indecomposable summands in a certain direct sum decomposition of a torsion-free abelian group of finite rank is very far from being an invariant of A.

Other surprising examples include torsion-free rank 2 groups A_n,m and B_n,m such that An is isomorphic to Bn if and only if n is divisible by m.

For abelian groups of infinite rank, there is an example of a group K and a subgroup G such that

• K is indecomposable;
• K is generated by G and a single other element; and
• Every nonzero direct summand of G is decomposable.