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### Questions (52)

 25 How can I prove that $xy\leq x^2+y^2$? 12 Prove that the additive group $ℚ$ is not isomorphic with the multiplicative group $ℚ^*$. 8 Let $σ∈S_n$ be a $n$ cycle. Prove that an element $τ∈S_n$ only commutes with $σ$ if $τ$ is a power of $σ$. 6 Let $G$ a group of order $6$. Prove that $G \cong \Bbb Z /6 \Bbb Z$ or $G \cong S_3$. 5 Consider on $\Bbb{R}$ the family $\Sigma$ of all Borel sets which are symmetric w.r.t. the origin. Show that $\Sigma$ is a $\sigma$-algebra.

### Reputation (932)

 +5 Let $G$ be a finite group and let $p$ a prime number that divides $|G|$. Let $h$ denote the number of subgroups of $G$ of order $p$. +5 Let $G$ a group of order $6$. Prove that $G \cong \Bbb Z /6 \Bbb Z$ or $G \cong S_3$. +5 Consider $f(z)=e^{-z^{-4}}$ for $z≠0$ and $f(0)=0$. Show that the Cauchy Riemann equations are satisfied for $z=0$ +5 How can I prove that $xy\leq x^2+y^2$?

 3 Find the taylor series for $\cos x$ and indicate why it converges to $\cos x$ for all $x\in \Bbb R$. 2 Does $|e^z| = e^x$ for $z = x + yi$? 1 About the epsilon definition of a convergent sequence. Is this definition equivalent?

### Tags (42)

 4 real-analysis × 11 1 definition 3 taylor-expansion × 4 1 education 2 complex-analysis × 5 0 ring-theory × 16 2 analysis 0 abstract-algebra × 15 1 convergence 0 group-theory × 8

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