A Belian group A that is the direct sum of cyclic groups is a group that is formed by taking the direct sum of several cyclic groups. This means that the group is composed of elements that can be written as tuples, where each element in the tuple belongs to a different cyclic group. The group operation is defined as the component-wise operation on the tuples.
The direct sum of cyclic groups is different from a direct product in that the elements in the direct sum are restricted to be tuples with only one non-zero element, while in a direct product, the tuples can have multiple non-zero elements. Additionally, the group operation in a direct product is defined as the component-wise operation, while in a direct sum, the operation is defined as the sum of the elements in the tuples.
A Belian group A that is the direct sum of cyclic groups can be represented as a direct sum of the individual cyclic groups that make up the group. Alternatively, it can also be represented as a direct product of the individual cyclic groups, with the additional restriction that the elements must be tuples with only one non-zero element.
Some examples of Belian groups A that are the direct sum of cyclic groups include the Klein four-group, the direct sum of two cyclic groups of order 2, and the direct sum of three cyclic groups of order 3. Additionally, any finite abelian group can be expressed as a direct sum of cyclic groups.
Belian groups A that are the direct sum of cyclic groups have many applications in mathematics, particularly in group theory and abstract algebra. They can be used to study and classify finite abelian groups, as well as to understand the structure of more complex groups. Additionally, they have applications in cryptography and coding theory.