In
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, a basic subgroup is a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
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 commu ...
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. As an example, the direct sum of two abelian groups A and B is anothe ...
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
Extension, extend or extended may refer to:
Mathematics
Logic or set theory
* Axiom of extensionality
* Extensible cardinal
* Extension (model theory)
* Extension (proof theory)
* Extension (predicate logic), the set of tuples of values t ...
between two well understood classes of abelian groups: direct sums of cyclic groups and
divisible group
In mathematics, specifically 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 positiv ...
s.
Definition and properties
A
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
, , 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 commu ...
, , 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 (mathematics), 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 ...
, , if the following conditions hold:
# is a direct sum of
cyclic group
In abstract algebra, a cyclic group or monogenous group is a Group (mathematics), group, denoted C_n (also frequently \Z_n or Z_n, not to be confused with the commutative ring of P-adic number, -adic numbers), that is Generating set of a group, ge ...
s of order and infinite cyclic groups;
# is a ''p''-
pure subgroup of ;
# The quotient group, , is a ''p''-
divisible group
In mathematics, specifically 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 positiv ...
.
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 ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
in . Picking a generator in each cyclic direct summand of creates a '' ''p''-basis'' of ''B'', which is analogous to a
basis
Basis is a term used in mathematics, finance, science, and other contexts to refer to foundational concepts, valuation measures, or organizational names; here, it may refer to:
Finance and accounting
* Adjusted basis, the net cost of an asse ...
of a
vector space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
or a
free abelian group
In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation (mathematics), operation that is associative, commutative, and inverti ...
.
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 divisibl ...
or ''bounded''; i.e., have bounded exponent. In general, the isomorphism class of the quotient, by a basic subgroup, , may depend on .
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