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Abstract Algebra – Sylow Theorems and Its applications

In abstract algebra, finite group theory is in its core. Now from Lagrange theorem, we have Lemma 1 Let be a subgroup of a finite group . Then the order of is a divisor of the order of . This theorem states the necessary condition for a finite group to be a subgroup of a…

December 2024
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  • Abstract Algebra – Sylow Theorems and Its applications

    In abstract algebra, finite group theory is in its core. Now from Lagrange theorem, we have Lemma 1 Let be a subgroup of a finite group . Then the order of is a divisor of the order of . This theorem states the necessary condition for a finite group to be a subgroup of a […]

    ../..

  • An approximation problem

    Problem: Let us call an irrational number well approximated by rational numbers if for any natural numbers there exists a rational number such that     Solution: 1. Pick     Why does it work ? Let’s say we let . Then we have     and     We now proceed in proving that […]

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  • On IMO2017 P2

    I did IMO 2017 P2 with my friends yesterday. Now I would like to share my thoughts on this problem. Problem: Let be the set of real numbers. Determine all functions such that, for any real numbers and ,     The initial step must be taking certain values, testing its properties and then start […]

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