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In [[Abstract algebra|abstract algebra]], a '''basic subgroup''' is a [[Subgroup|subgroup]] of an [[Abelian group|abelian group]] which is a [[Direct sum|direct sum]] of [[Cyclic group|cyclic subgroups]] and satisfies further technical conditions. This notion was introduced by L. Ya. Kulikov (for [[P-group|''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 [[Group extension|extensions]] between two well understood classes of abelian groups: direct sums of cyclic groups and [[Divisible group|divisible group]]s.
== Definition and properties ==

A [[Subgroup|subgroup]], {{math|''B''}}, of an [[Abelian group|abelian group]], {{math|''A''}}, is called ''' ''p''-basic''', for a fixed [[Prime number|prime number]], {{math|''p''}}, if the following conditions hold:

# {{math|''B''}} is a direct sum of [[Cyclic group|cyclic group]]s of order {{math|''p{{sup|n}}''}} and infinite cyclic groups;
# {{math|''B''}} is a ''p''-[[Pure subgroup|pure subgroup]] of {{math|''A''}};
# The quotient group, {{math|''A''/''B''}}, is a ''p''-[[Divisible group|divisible group]].

Conditions 1–3 imply that the subgroup, {{math|''B''}}, is Hausdorff in the ''p''-adic topology of {{math|''B''}}, which moreover coincides with the topology [[Induced topology|induced]] from {{math|''A''}}, and that {{math|''B''}} is dense in {{math|''A''}}. Picking a generator in each cyclic direct summand of {{math|''B''}} creates a '' ''p''-basis'' of ''B'', which is analogous to a [[Basis (linear algebra)|basis]] of a [[Vector space|vector space]] or a [[Free abelian group|free abelian group]].

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

== 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|principal ideal domain]]. The existence of such a ''basic submodule'' and uniqueness of its isomorphism type continue to hold.{{Citation needed|reason=Reliable source needed for the whole sentence|date=July 2017}}

== References ==

* László Fuchs (1970), ''Infinite abelian groups, Vol. I''. Pure and Applied Mathematics, Vol. 36. New York–London: Academic Press {{MR|0255673}}
* L. Ya. Kulikov, ''On the theory of abelian groups of arbitrary cardinality'' (in Russian), Mat. Sb., 16 (1945), 129–162
*{{Citation | last1=Kurosh | first1=A. G. | title=The theory of groups | publisher=Chelsea | location=New York | mr=0109842 | year=1960}}

[[Category:Abelian group theory]]
[[Category:Infinite group theory]]
[[Category:Subgroup properties]]

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