
Direct sum of groups

In mathematics, a group "G" is called the direct sum of two subgroups "H"1 and "H"2 if
More generally, "G" is called the direct sum of a finite set of subgroups {"H""i"} if
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 {"H""i"} then we often write "G" = ∑"H""i". Loosely speaking, a direct sum is isomorphic to a weak direct product of subgroups.
In abstract algebra, this method of construction can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information.
This direct sum is commutative up to isomorphism. That is, if then also and thus . It is also associative in the sense that if , and , then .
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".
If , then it can be proven that:
The above assertions can be generalized to the case of , where {"H"i} is a finite set of subgroups:
Note the similarity with the direct product, where each "g" can be expressed uniquely as
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, ∑"H""i" is isomorphic to the direct product ×{"H""i"}.
Direct summand.
Given a group formula_2, we say that a subgroup formula_3 is a direct summand of formula_2 if there exists another subgroup formula_5 of formula_2 such that formula_7.
In abelian groups, if formula_3 is a divisible subgroup of formula_2, then formula_3 is a direct summand of formula_2.
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 formula_25 we have that
However, the Remak-Krull-Schmidt theorem states that given a "finite" group "G" = ∑"A""i" = ∑"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 ∏{"H""i"} 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 {"H""i"} (written as ∑E{"H""i"}) is the subset of ∏{"H""i"}, where, for each element "g" of ∑E{"H""i"}, "g""i" is the identity formula_28 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 {"H""i"} the external direct sum is equal to the direct product.
If "G" = ∑"H""i", then "G" is isomorphic to ∑E{"H""i"}. 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 {"h""i" ∈ "H""i" : "i" ∈ "S"} such that "g" = ∏ {"h""i" : "i" in "S"}.