N. S.
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 10h awarded convergence 13h comment Only two groups of order $10$: $C_{10}$ and $D_{10}$ ... The last is impossible: in that case the group would be Abelian (standard exercise in any introductory book) and it is not too hard to see that it is impossible for an Abelian group to have 10 elements, and all elements to have order 1 or 2... 13h comment Only two groups of order $10$: $C_{10}$ and $D_{10}$ @IrregularUser In this case one can prove the existence of those elements at an elementary level, no Cauchy Theorem is needed. By Lagrange every element has order 1,2,5 or 10, and there is an element of order 1. If there is an element of order 10 we are done. Otehrwise every element has order 2 or 5.... The elements of order 5 can be grouped in groups of $4$ (as together with $e$ they generated a 5-group). Thus it is impossible for the remaining $9$ elements to all have order $5$. So either there is an element of order 5 and one element of order 2, or all elements have order 2... 13h comment Only two groups of order $10$: $C_{10}$ and $D_{10}$ 19h comment Prove $a_t \rightarrow x$ using the Betweenness Property @HagenvonEitzen Nice, I was thinking about picking it increasing intuitively, your hint is much nicer :) 19h answered Prove $a_t \rightarrow x$ using the Betweenness Property 1d comment If an R-module P is free and A and B are direct summands of P then A∩B is isomorphic to a direct summand of P? is it true? I could not prove it? If I am not mistaken, your question is exactly this one: math.stackexchange.com/questions/177806/… 2d answered If $z_0$ is a root of the equation $z^n\cos\theta_0+z^{n-1}\cos\theta_1+\cdots+\cos\theta_n=2$ 2d comment Is $0$ a natural number? generally speaking 2 is unlike the other primes, it is special among primes in soo many different ways.... Should we exclude him fro the list of primes? 2d comment Is $0$ a natural number? @celtschk If you define rational numbers that way, you miss the number 0 ;) So you don;t define all rational numbers..... 2d answered Proof for $log\left(\sum_{n=1}^{\infty} \frac{1}{n}\right)$ diverging. 2d answered Can we say that $\det(A+B) = \det(A) + \det(B) +\operatorname{tr}(A) \operatorname{tr}(B) - \operatorname{tr}(AB)$. 2d comment $\int\frac{\sin(nx) \sin x}{1-\cos x} \,dx$ by elementary methods @MarkFischler One more comment: After the suggested change,the identity you need to prove becomes just some identity for the Chebyshev polynomials, and might be easier to prove that way. 2d comment $\int\frac{\sin(nx) \sin x}{1-\cos x} \,dx$ by elementary methods @MarkFischler Check the complex solution I just fixed, I think it is faster. It also suggests that if you try to do it by induction, it might be easier to write $\frac{\cos((n-1)x)-\cos((n+1)x))}{2}$ instead of $\sin(nx)\sin(x)$. 2d comment $\int\frac{\sin(nx) \sin x}{1-\cos x} \,dx$ by elementary methods @JeanMarie Good point, fixed it. 2d revised $\int\frac{\sin(nx) \sin x}{1-\cos x} \,dx$ by elementary methods added 515 characters in body 2d comment Number of solutions for the given equation in finite field of order 32. @LavKumar What about now? 2d revised Number of solutions for the given equation in finite field of order 32. added 210 characters in body 2d answered Number of solutions for the given equation in finite field of order 32. Feb 4 answered Find the polynomial $P$ of smallest degree with rational coefficients and leading coefficient $1$ such that $P(49^{1/3}+7^{1/3})=4$