Chiara
Tue Sep 17 2024
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5 answers
975
Does abelian mean cyclic?
I'm curious to know, does the term 'abelian' inherently mean that a group is also 'cyclic'? I understand that abelian groups have the commutative property, where the order of multiplication doesn't matter. But does this automatically imply that every element in the group can be generated by a single element, which is the defining characteristic of a cyclic group? Or are there abelian groups that are not cyclic? I'd appreciate a clear explanation to help me better grasp this concept in group theory.
Gianluca
Sun Aug 25 2024
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5 answers
1013
Why is the Klein 4 group not cyclic?
Can you explain to me, in simple terms, why the Klein 4 group, also known as the Vierergruppe or the group of order 4, is not considered cyclic? I understand that a cyclic group is one in which the elements are arranged in a way that every non-identity element can be expressed as a power of a single element, known as the generator. So, what specific property or characteristic of the Klein 4 group prevents it from being cyclic?
