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There is a cool problem from Herstein’s Topics in Algebra. It states Let be a finite group whose order is not divisible by 3. Suppose that for all for all . Prove that must be abelian. I recall solving this … Continue reading
Here is a problem that appeared in Romanian Mathematical Olympiad (Junior Team Selection Test 2002). If prove that First Solution: Use CauchySchwarz inequality to obtain: Here we used the fact that (which follows from ), and (which follows from ). … Continue reading
Today I was reading some basic group theory from Herstein’s Topics in Algebra, and saw the following cute problem: Prove that every group of order 5 is abelian. Now, clearly Lagrange’s theorem implies that there is only one group of … Continue reading
One of the things that baffled me when I was learning calculus was that harmonic series diverges. Clearly, the terms are getting smaller and smaller (and in fact the terms go to zero in the limit the number of terms … Continue reading