### Questions (138)

 7 Differentiability of $G(x)=\int_{\mathbb R} e^{tx}f(t)dt$ on $(0,1)$ 7 Suppose $\alpha, \beta>0$. Compute: $\int_{0}^{\infty}\frac{\cos (\alpha x)-\cos (\beta x)}{x}dx$ 6 What is the intuition behind the law of quadratic reciprocity? 6 Suppose that $G$ is a group of order $924=2^2\cdot3\cdot7\cdot 11$. Prove that $G$ has an element of order $77$. 5 Real roots of $1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots +\frac{x^n}{n!}$ [duplicate]

### Reputation (4,452)

 +10 How to verify that an $\mathbb F[G]$-module has an imprimitivity decomposition or not? +10 Order of elements not in any Sylow $2$-subgroup +10 Any cyclic submodule $Dz$ such that $\operatorname{ann} z\subset \operatorname{ann} x$ for every $x \in M$, is a pure submodule. +10 Abelian subgroup of a group of order $2002$

 6 Evaluate the limit the following series 6 Evaluate $\sum_{k=0}^n {{2n + 1}\choose {2k + 1}}$ 6 Finding all the ideals of $\mathbb{Z} [x]/(2, (x^3+1))$? 5 What is the difference between $\Bbb F_4$ and $\Bbb Z/4\Bbb Z$ 4 Classifying all non-abelian groups of order $55$

### Tags (140)

 51 abstract-algebra × 65 11 limits × 10 18 group-theory × 26 10 combinatorics × 6 14 ring-theory × 8 10 finite-fields × 6 13 real-analysis × 42 9 sequences-and-series × 6 11 calculus × 18 9 discrete-mathematics × 2

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