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In mathematics, a group G is called the direct sum [1] [2] of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules for more information.
For an arbitrary family of groups indexed by , their direct sum [2] is the subgroup of the direct product that consists of the elements () that have finite support, where by definition, () is said to have finite support if is the identity element of for all but finitely many . [3] The direct sum of an infinite family () of non-trivial groups is ...
In particular, the direct sum of square matrices is a block diagonal matrix. The adjacency matrix of the union of disjoint graphs (or multigraphs) is the direct sum of their adjacency matrices. Any element in the direct sum of two vector spaces of matrices can be represented as a direct sum of two matrices. In general, the direct sum of n ...
More generally, is called the direct sum of a finite set of subgroups, …, of the map = is a topological isomorphism. If a topological group G {\displaystyle G} is the topological direct sum of the family of subgroups H 1 , … , H n {\displaystyle H_{1},\ldots ,H_{n}} then in particular, as an abstract group (without topology) it is also the ...
Direct sums are commutative and associative (up to isomorphism), meaning that it doesn't matter in which order one forms the direct sum. The abelian group of R-linear homomorphisms from the direct sum to some left R-module L is naturally isomorphic to the direct product of the abelian groups of R-linear homomorphisms from M i to L: (,) (,).
As well as the direct sum, another way to combine free abelian groups is to use the tensor product of -modules. The tensor product of two free abelian groups is always free abelian, with a basis that is the Cartesian product of the bases for the two groups in the product. [22]
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A representation is called semisimple or completely reducible if it can be written as a direct sum of irreducible representations. This is analogous to the corresponding definition for a semisimple algebra. For the definition of the direct sum of representations please refer to the section on direct sums of representations.