Direct sum of groups
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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a group ''G'' is called the direct sumHomology. 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, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
, 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 (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called sc ...
s, modules, 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, m ...
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 ''G'' is called the direct sum of two
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  ...
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 is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
e 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, 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  ...
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 ...
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 or morphism 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 the ...
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, 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 G, we say that a subgroup H is a direct summand of G if there exists another subgroup K of G such that G = H+K. In abelian groups, if H is a divisible subgroup of G, then H is a direct summand of G.


Examples

* If we take G= \prod_ H_i it is clear that G is the direct product of the subgroups H_ \times \prod_H_i. * If H is a divisible subgroup of an abelian group G then there exists another subgroup K of G such that G=K+H. * If G also has 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 ...
structure then G can be written as a direct sum of \mathbb R and another subspace K that will be isomorphic to the quotient G/K.


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 V_4 \cong C_2 \times C_2 we have that : V_4 = \langle(0,1)\rangle + \langle(1,0)\rangle, and : V_4 = \langle(1,1)\rangle + \langle(1,0)\rangle. 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 and , denoted , is the set of all ordered pairs where is an element of and is an element of . In terms of set-builder notation, that is A\times B = \. A table c ...
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 e_ 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. As an example, the direct sum of two abelian groups A and B is anothe ...
*
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 cop ...
* Free product * Direct sum of topological groups


References

{{DEFAULTSORT:Direct Sum Of Groups Group theory