Arianna
Mon Aug 12 2024
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6 answers
968
Why is S4 not abelian?
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?
GeishaMelody
Mon Aug 12 2024
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5 answers
676
Why is d4 not abelian?
Could you please elaborate on why the group d4, often referred to as the dihedral group of order 8, is not considered abelian? It's my understanding that in an abelian group, the order of multiplication doesn't matter, i.e., for any two elements a and b in the group, a * b = b * a. However, with d4, which comprises rotations and reflections of a square, it seems the order in which we apply these transformations can yield different results. Could you explain why this property disqualifies d4 from being abelian?
Riccardo
Mon Aug 12 2024
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5 answers
1478
Is every abelian group solvable?
I'm curious to know, does the concept of solvability apply to all abelian groups? Considering that abelian groups possess a certain level of symmetry and simplicity in their structure, does this inherently mean that they can always be decomposed into simpler subgroups in a finite number of steps? Or are there specific conditions or properties that an abelian group must possess in order to be classified as solvable? I'm eager to understand the nuances and implications of this question within the realm of group theory and its applications to cryptography, algebra, and other fields of mathematics.
