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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 = ΣH i, then G is isomorphic to Σ E {H i}. Thus, in a sense, the direct sum is an "internal ...
In the category of rings, the coproduct is given by a construction similar to the free product of groups.) Use of direct sum terminology and notation is especially problematic when dealing with infinite families of rings: If () is an infinite collection of nontrivial rings, then the direct sum of the underlying additive groups can be equipped ...
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.
The concept of disjoint union secretly underlies the above examples: the direct sum of abelian groups is the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero), similarly for vector spaces: the space spanned by the "almost" disjoint union; the free product for groups is generated ...
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 ...
In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, in the ...
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