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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 ...
The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups. [13]
General case: [2] In category theory the direct sum is often, but not always, the coproduct in the category of the mathematical objects in question. For example, in the category of abelian groups, direct sum is a coproduct. This is also true in the category of modules.
In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted . Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups , every finite abelian group can be expressed as the direct sum of cyclic groups .
The structure of any finite abelian group is relatively simple; every finite abelian group is the direct sum of cyclic p-groups. This can be extended to a complete classification of all finitely generated abelian groups, that is all abelian groups that are generated by a finite set. The situation is much more complicated for the non-abelian groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra.
In abstract algebra, a basic subgroup is a subgroup of an abelian group which is a direct sum of cyclic subgroups and satisfies further technical conditions. This notion was introduced by L. Ya. Kulikov (for p-groups) and by László Fuchs (in general) in an attempt to formulate classification theory of infinite abelian groups that goes beyond the Prüfer theorems.
An object in Ab is injective if and only if it is a divisible group; it is projective if and only if it is a free abelian group. The category has a projective generator (Z) and an injective cogenerator (Q/Z). Given two abelian groups A and B, their tensor product A⊗B is defined; it is again an abelian group.