abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The te ...

, a basic subgroup is a

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...

of an

abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is com ...

which is a

direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct sum is used in abstract algebra, consider a mo ...

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 In mathematics, two Prüfer theorems, named after Heinz Prüfer, describe the structure of certain infinite abelian groups. They have been generalized by L. Ya. Kulikov. Statement Let ''A'' be an abelian group. If ''A'' is finitely generated t ...

. It helps to reduce the classification problem to classification of possible

extensions Extension, extend or extended may refer to: Mathematics Logic or set theory * Axiom of extensionality * Extensible cardinal * Extension (model theory) * Extension (predicate logic), the set of tuples of values that satisfy the predicate * Ext ...

between two well understood classes of abelian groups: direct sums of cyclic groups and

divisible group In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an ''n''th multiple for each positive i ...

Definition and properties

, , of an

, , is called ''p''-basic, for a fixed

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...

, , if the following conditions hold: # is a direct sum of

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...

s of order and infinite cyclic groups; # is a ''p''- pure subgroup of ; # The quotient group, , is a ''p''-

. Conditions 1–3 imply that the subgroup, , is Hausdorff in the ''p''-adic topology of , which moreover coincides with the topology induced from , and that is

dense Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...

in . Picking a generator in each cyclic direct summand of creates a '' ''p''-basis'' of ''B'', which is analogous to a

basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting o ...

of a

vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...

or a

free abelian group In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a su ...

. Every abelian group, , contains ''p''-basic subgroups for each , and any 2 ''p''-basic subgroups of 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 In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

or ''bounded''; i.e., have bounded exponent. In general, the isomorphism class of the quotient, by a basic subgroup, , may depend on .

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 In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are princip ...

. The existence of such a ''basic submodule'' and uniqueness of its isomorphism type continue to hold.

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 *{{Citation , last1=Kurosh , first1=A. G. , authorlink=Aleksandr Gennadievich Kurosh , title=The theory of groups , publisher=Chelsea , location=New York , mr=0109842 , year=1960 Abelian group theory Infinite group theory Subgroup properties