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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 ...
3. Between two groups, may mean that the first one is a proper subgroup of the second one. > (greater-than sign) 1. Strict inequality between two numbers; means and is read as "greater than". 2. Commonly used for denoting any strict order. 3. Between two groups, may mean that the second one is a proper subgroup of the first one. ≤ 1.
More generally, is called the direct sum of a finite set of subgroups, …, of the map = is a topological isomorphism. If a topological group is the topological direct sum of the family of subgroups , …, then in particular, as an abstract group (without topology) it is also the direct sum (in the usual way) of the family .
The direct sum and direct product are not isomorphic for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of category theory: the direct sum is the coproduct, while the direct product is the product.
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: (,) (,).
A module is called torsionless if it embeds into its algebraic dual. Simple A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible. [5] Semisimple A semisimple module is a direct sum (finite or not) of simple modules.
The pushout of these maps is the direct sum of A and B. Generalizing to the case where f and g are arbitrary homomorphisms from a common domain Z, one obtains for the pushout a quotient group of the direct sum; namely, we mod out by the subgroup consisting of pairs (f(z), −g(z)). Thus we have "glued" along the images of Z under f and g.