Cryptocurrency Q&A
What is an example of an abelian group?
What is an example of an abelian group?
BonsaiBeauty
Wed Sep 18 2024
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5 answers
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Could you provide an example of an abelian group, perhaps with a clear explanation of its properties? It would be helpful to understand how the elements of this group interact with each other under the operation defined, and why this group is considered to be abelian in nature. By breaking it down in this way, it may be easier for those who are new to the concept to grasp the fundamental principles of abelian groups.
5 answers
Eleonora
Fri Sep 20 2024
This uniqueness signifies that any abelian group of order 15 possesses an identical algebraic structure, making them isomorphic to each other. Isomorphism, in mathematics, refers to a relationship between two structures where one structure can be mapped onto the other in a way that preserves the structure's properties.
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GangnamGlitzGlamourGloryDays
Fri Sep 20 2024
To illustrate this concept, consider a concrete example of an abelian group of order 15. It can be constructed by taking the direct sum of two subsets: {0, 5, 10} and {0, 3, 6, 9, 12}. These subsets themselves form cyclic groups under addition modulo their respective cardinalities.
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Caterina
Fri Sep 20 2024
The first subset, {0, 5, 10}, is a cyclic group of order 3, as adding any element to itself twice results in the identity element (0). Similarly, the second subset, {0, 3, 6, 9, 12}, is a cyclic group of order 5.
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Lorenzo
Fri Sep 20 2024
Abelian groups of a specific order, such as 15, exhibit a unique decomposition. In the case of order 15, the group can solely be expressed as the direct sum of two cyclic groups, Z3 and Z5.
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HallyuHeroLegendaryStar
Fri Sep 20 2024
By combining these two cyclic groups through the direct sum operation, we obtain an abelian group of order 15 that encompasses all possible combinations of elements from both subsets. This construction underscores the fact that all abelian groups of order 15 share a common algebraic blueprint.
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