Английская Википедия:Basic subgroup

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In abstract algebra, a basic subgroup is a subgroup of an abelian group which is a direct sum of cyclic subgroups and satisfies further technical conditions. This notion was introduced by L. Ya. Kulikov (for p-groups) and by László Fuchs (in general) in an attempt to formulate classification theory of infinite abelian groups that goes beyond the Prüfer theorems. It helps to reduce the classification problem to classification of possible extensions between two well understood classes of abelian groups: direct sums of cyclic groups and divisible groups.

Definition and properties

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

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

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

Every abelian group, Шаблон:Math, contains p-basic subgroups for each Шаблон:Math, and any 2 p-basic subgroups of Шаблон:Math 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, Шаблон:Math by a basic subgroup, Шаблон:Math, may depend on Шаблон:Math.

Generalization to modules

The notion of a p-basic subgroup in an abelian p-group admits a direct generalization to modules over a principal ideal domain. The existence of such a basic submodule and uniqueness of its isomorphism type continue to hold.Шаблон:Citation needed

References

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