Concept

Direct sum of groups

Résumé
In mathematics, a group G is called the direct sum 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. A group which can be expressed as a direct sum of non-trivial subgroups is called decomposable, and if a group cannot be expressed as such a direct sum then it is called indecomposable. Definition A group G is called the direct sum of two subgroups H1 and H2 if
• each H1 and H2 are normal subgroups of G,
• the subgroups H1 and H2 have trivial intersection (i.e., having only the identity element e of G in common),
• G = ⟨H1, H2⟩; in other words, G is generated by the subgroups H1 and H2.
More generally, G is called the direct sum of a finite set of subgroups {Hi} if
• each Hi is a normal sub
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