Cryptocurrency Q&A
Does abelian mean cyclic?
Does abelian mean cyclic?
Chiara
Tue Sep 17 2024
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5 answers
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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.
5 answers
EnchantedNebula
Thu Sep 19 2024
Cyclic groups, characterized by the presence of a single element that generates the entire group through its powers, are inherently Abelian in nature. This property stems from the commutative behavior of their elements under the group operation.
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SamsungShineBrightness
Wed Sep 18 2024
The character table of an Abelian group encapsulates the essence of its representation theory. A notable aspect of this table is that it revolves around the powers of a single element, known as the group generator, in cyclic groups. However, in general Abelian groups, the character table reflects the group's overall structure through the contributions of all its elements, albeit in a simplified form due to the Abelian property.
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GyeongjuGloryDaysFestival
Wed Sep 18 2024
Conversely, while all cyclic groups are Abelian, not every Abelian group can be classified as cyclic. Abelian groups possess a broader structure, allowing for more complex compositions beyond the simple repetitive pattern of cyclic groups.
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emma_grayson_journalist
Wed Sep 18 2024
A salient feature of Abelian groups is that all their subgroups are necessarily normal. This normality arises from the commutative property, ensuring that left and right cosets coincide, thereby satisfying the definition of normal subgroups.
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noah_doe_writer
Wed Sep 18 2024
In the context of Abelian groups, each element forms a distinct conjugacy class by itself. This is a direct consequence of the group's commutative property, where every element commutes with every other element, eliminating the possibility of non-singleton conjugacy classes.
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