Cryptocurrency Q&A
Is every abelian group solvable?
Is every abelian group solvable?
Riccardo
Mon Aug 12 2024
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5 answers
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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.
5 answers
Martina
Wed Aug 14 2024
The solvability of nilpotent groups is another notable aspect. Nilpotency, a property related to the length of the upper central series, ensures that such groups are also solvable.
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SakuraSpiritual
Wed Aug 14 2024
The direct product of solvable groups, when finite, retains the solvability property. This means that if we have a finite collection of solvable groups and take their direct product, the resulting group is still solvable.
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CryptoAce
Wed Aug 14 2024
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Silvia
Wed Aug 14 2024
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Bianca
Wed Aug 14 2024
Abelian groups, characterized by their commutative property, are inherently solvable. This stems from the fact that any abelian group G can be expressed as a solvable series, specifically G = H0 ⊇ H1 = {e}, where H1 is the trivial subgroup containing only the identity element e.
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