TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a
group A group is a number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
''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 algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), rings, field (mathema ...
, this method of construction of groups can be generalized to direct sums of
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 syst ...
, and other structures; see the article
direct sum of modules In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...
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 A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...
''G'' is called the direct sum of two
subgroup In group theory In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
$e$ 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ...
s if * each ''H''''i'' is a
normal subgroup In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...
, 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
$V_4 \cong C_2 \times C_2$ we have that : $V_4 = \langle\left(0,1\right)\rangle + \langle\left(1,0\right)\rangle,$ and : $V_4 = \langle\left(1,1\right)\rangle + \langle\left(1,0\right)\rangle.$ 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 .