Basic subgroup

Definition and properties

A subgroup, B, of an abelian group, A, is called ** p-basic**, for a fixed prime number, p, if the following conditions hold:

1. B is a direct sum of cyclic groups of order p and infinite cyclic groups;
2. B is a p-pure subgroup of A;
3. The quotient group, A/B, is a p-divisible group.

Conditions 1–3 imply that the subgroup, B, is Hausdorff in the p-adic topology of B, which moreover coincides with the topology induced from A, and that B is dense in A. Picking a generator in each cyclic direct summand of B creates a * p-basis* of B, which is analogous to a basis of a vector space or a free abelian group.

Every abelian group, A, contains p-basic subgroups for each p, and any 2 p-basic subgroups of A are isomorphic. Abelian groups that contain a unique p-basic subgroup have been completely characterized. For the case of p-groups they are either divisible or bounded; i.e., have bounded exponent. In general, the isomorphism class of the quotient, A/B by a basic subgroup, B, may depend on B.

References

• László Fuchs (1970), Infinite abelian groups, Vol. I. Pure and Applied Mathematics, Vol. 36. New York–London: Academic Press
• L. Ya. Kulikov, On the theory of abelian groups of arbitrary cardinality (in Russian), Mat. Sb., 16 (1945), 129–162