In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
''G'' is called the direct sum
[Homology. Saunders MacLane. Springer, Berlin; Academic Press, New York, 1963.][László Fuchs. Infinite Abelian Groups] of two
normal subgroups with
trivial intersection if it is
generated by the subgroups. In
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 term ''a ...
, this method of construction of groups can be generalized to direct sums of
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 can ...
s,
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
, and other structures; see the article
direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, ma ...
for more information. A group which can be expressed as a direct sum of non-trivial subgroups is called ''decomposable'', and if a group cannot be expressed as such a direct sum then it is called ''indecomposable''.
Definition
A
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
''G'' is called the direct sum
of two
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 subgroup ...
s ''H''
1 and ''H''
2 if
* each ''H''
1 and ''H''
2 are normal subgroups of ''G'',
* the subgroups ''H''
1 and ''H''
2 have trivial intersection (i.e., having only the
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
of ''G'' in common),
* ''G'' = ⟨''H''
1, ''H''
2⟩; in other words, ''G'' is generated by the subgroups ''H''
1 and ''H''
2.
More generally, ''G'' is called the direct sum of a finite set of
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 subgroup ...
s if
* each ''H''
''i'' is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of ''G'',
* each ''H''
''i'' has trivial intersection with the subgroup ,
* ''G'' = ⟨⟩; in other words, ''G'' is
generated by the subgroups .
If ''G'' is the direct sum of subgroups ''H'' and ''K'' then we write , and if ''G'' is the direct sum of a set of subgroups then we often write ''G'' = Σ''H''
''i''. Loosely speaking, a direct sum is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to a weak direct product of subgroups.
Properties
If , then it can be proven that:
* for all ''h'' in ''H'', ''k'' in ''K'', we have that
* for all ''g'' in ''G'', there exists unique ''h'' in ''H'', ''k'' in ''K'' such that
* There is a cancellation of the sum in a quotient; so that is isomorphic to ''H''
The above assertions can be generalized to the case of , where is a finite set of subgroups:
* if , then for all ''h''
''i'' in ''H''
''i'', ''h''
''j'' in ''H''
''j'', we have that
* for each ''g'' in ''G'', there exists a unique set of elements ''h''
''i'' in ''H''
''i'' such that
:''g'' = ''h''
1 ∗ ''h''
2 ∗ ... ∗ ''h''
''i'' ∗ ... ∗ ''h''
''n''
* There is a cancellation of the sum in a quotient; so that is isomorphic to Σ''H''
''i''.
Note the similarity with the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
, where each ''g'' can be expressed uniquely as
:''g'' = (''h''
1,''h''
2, ..., ''h''
''i'', ..., ''h''
''n'').
Since for all , it follows that multiplication of elements in a direct sum is isomorphic to multiplication of the corresponding elements in the direct product; thus for finite sets of subgroups, Σ''H''
''i'' is isomorphic to the direct product ×.
Direct summand
Given a group
, we say that a subgroup
is a direct summand of
if there exists another subgroup
of
such that
.
In abelian groups, if
is a
divisible subgroup of
, then
is a direct summand of
.
Examples
* If we take
it is clear that
is the direct product of the subgroups
.
* If
is a
divisible subgroup of an abelian group
then there exists another subgroup
of
such that
.
* If
also has 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 can ...
structure then
can be written as a direct sum of
and another subspace
that will be isomorphic to the quotient
.
Equivalence of decompositions into direct sums
In the decomposition of a finite group into a direct sum of indecomposable subgroups the embedding of the subgroups is not unique. For example, in the
Klein group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one. ...
we have that
:
and
:
However, the
Remak-Krull-Schmidt theorem states that given a ''finite'' group ''G'' = Σ''A''
''i'' = Σ''B''
''j'', where each ''A''
''i'' and each ''B''
''j'' is non-trivial and indecomposable, the two sums have equal terms up to reordering and isomorphism.
The Remak-Krull-Schmidt theorem fails for infinite groups; so in the case of infinite ''G'' = ''H'' + ''K'' = ''L'' + ''M'', even when all subgroups are non-trivial and indecomposable, we cannot conclude that ''H'' is isomorphic to either ''L'' or ''M''.
Generalization to sums over infinite sets
To describe the above properties in the case where ''G'' is the direct sum of an infinite (perhaps uncountable) set of subgroups, more care is needed.
If ''g'' is an element of the
cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\ti ...
Π of a set of groups, let ''g''
''i'' be the ''i''th element of ''g'' in the product. The external direct sum of a set of groups (written as Σ
''E'') is the subset of Π, where, for each element ''g'' of Σ
''E'', ''g''
''i'' is the identity
for all but a finite number of ''g''
''i'' (equivalently, only a finite number of ''g''
''i'' are not the identity). The group operation in the external direct sum is pointwise multiplication, as in the usual direct product.
This subset does indeed form a group, and for a finite set of groups the external direct sum is equal to the direct product.
If ''G'' = Σ''H''
''i'', then ''G'' is isomorphic to Σ
''E''. Thus, in a sense, the direct sum is an "internal" external direct sum. For each element ''g'' in ''G'', there is a unique finite set ''S'' and a unique set such that ''g'' = Π .
See also
*
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 more ...
*
Coproduct
In category theory, the coproduct, or categorical sum, is a construction which includes as examples the disjoint union of sets and of topological spaces, the free product of groups, and the direct sum of modules and vector spaces. The coprodu ...
*
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, and i ...
*
Direct sum of topological groups In mathematics, a topological group G is called the topological direct sum of two subgroups H_1 and H_2 if the map
\begin
H_1\times H_2 &\longrightarrow G \\
(h_1,h_2) &\longmapsto h_1 h_2
\end
is a topological isomorphism, meaning that ...
References
{{DEFAULTSORT:Direct Sum Of Groups
Group theory