Cryptocurrency Q&A
Why is S4 not abelian?
Why is S4 not abelian?
Arianna
Mon Aug 12 2024
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6 answers
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I'm curious to understand why the group S4, which represents the set of all permutations of four distinct elements, is not abelian. Can you explain the underlying mathematical principles that lead to this non-abelian property? Specifically, what are the key differences in the behavior of elements within S4 compared to abelian groups, and how do these differences manifest in terms of the group's operation?
6 answers
Claudio
Wed Aug 14 2024
Specifically, if we denote the normal subgroup of S4 by N, the quotient group S4/N inherits certain traits from its parent group. Given that S4 lacks elements of order 6, a logical deduction emerges: S4/N too, must be devoid of such elements.
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Martino
Wed Aug 14 2024
This crucial observation leads us to a remarkable conclusion. Since S4/N cannot possess elements of order 6, it must necessarily be isomorphic to another well-known group—S3. This isomorphism underscores the deep connections and shared properties between these two groups.
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Rosalia
Wed Aug 14 2024
It is worth noting that S3, being a non-abelian group, possesses a unique structure that distinguishes it from abelian groups. The fact that S4/N is equivalent to S3 underscores the non-abelian nature of the quotient group, further highlighting its complexity and significance.
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EnchantedDreams
Wed Aug 14 2024
In the realm of abstract algebra, the concept of group order is paramount. When discussing groups of order 6, we encounter two distinct yet significant entities: S3 and C6. These two groups are the sole representatives of this order, each with its unique properties and characteristics.
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DigitalLord
Wed Aug 14 2024
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