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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 mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H.This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.
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 ...
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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If a character of the finite group G is restricted to a subgroup H, then the result is also a character of H. Every character value χ(g) is a sum of n m-th roots of unity, where n is the degree (that is, the dimension of the associated vector space) of the representation with character χ and m is the order of g.