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How to prove something is abelian?
How to prove something is abelian?
KpopHarmony
Thu Aug 15 2024
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7 answers
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Hello, I'm curious about the process of proving whether a mathematical object is abelian. Could you explain in simple terms what an abelian group is, and then outline the general steps one might take to demonstrate that a particular group possesses this property? Additionally, are there any common pitfalls or misconceptions that one should be aware of when approaching this type of proof? Thank you for your time and expertise.
7 answers
Sara
Fri Aug 16 2024
A direct product of groups combines two or more groups into a larger group, where each element of the larger group can be uniquely expressed as a tuple of elements from the smaller groups. In the context of Abelian groups, this implies that the combined group inherits the commutative property from its constituent Abelian subgroups.
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Giulia
Fri Aug 16 2024
To demonstrate that a group is Abelian, one must prove that the commutator of any two arbitrary elements within the group equals the identity element. The commutator, denoted as [x,y], is defined as the product of x and y followed by the inverse of y and then the inverse of x, subtracted from the identity.
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Claudio
Fri Aug 16 2024
Specifically, the commutator [x,y] = xyx−1y−1 must evaluate to the identity element for all x,y belonging to the group G. This condition ensures that the order of multiplication of elements within the group does not affect the final result, a defining characteristic of Abelian groups.
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BitcoinBaroness
Fri Aug 16 2024
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Lucia
Fri Aug 16 2024
Abelian groups possess a unique property where the group operation is commutative, meaning that for any two elements a and b in the group, the result of the group operation on a and b is the same as the result of the group operation on b and a.
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