# Tagged Questions

For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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### every Abelian group is a converse lagrange theorem group

Let $G$ be a finite abelian group, then $G$ has a subgroup of order $n$ if and only if $n\mid G$. Proof: by Lagrange if $H\leq G$ then $|H|$ divides $|G|$ so this proves one of the implications. We ...
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### If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$ [duplicate]

If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$ So far I've got: $a<b$ $a^2<ba$ $a<\sqrt{ab}$ And: $a<b$ $a+b<2b$ $\frac{a+b}{2}<b$ So I need to prove ...
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### If $f$ continuous and $\lim_{x\to-\infty }f(x)=\lim_{x\to\infty }f(x)=+\infty$ then $f$ takes its minimum.

Let $f:\mathbb R\to\mathbb R$ continuous and $$\lim_{x\to-\infty }f(x)=\lim_{x\to\infty }f(x)=+\infty$$ then $f$ takes its minimum. This is a homework. My solution looks to simple, that's why I would ...
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### Proving that there are infinitely many primes with remainder of 2 when divided by 3

I need to prove that there are infinitely many primes with remainder of 2 when divided by 3. I started out similarly to Euclid's classic proof of an infinite number of prime numbers: Suppose there is ...
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### Prove that the additive groups $\mathbb{Z}$ and $\mathbb{Q}$ are not isomorphic.

Is my proof below correct? What specific property of rationals did I exploit in my proof? It looks like the property I exploited is the following: Given any positive rational, I can always write it as ...
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### Is a matrix $A$ with an eigenvalue of $0$ invertible?

Just wanted some input to see if my proof is satisfactory or if it needs some cleaning up. Here is what I have. Proof Suppose $A$ is square matrix and invertible and, for the sake of ...
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### Prove that the multiplicative groups $\mathbb{R} - \{0\}$ and $\mathbb{C} - \{0\}$ are not isomorphic.

Is my proof correct? I have made use of the fact isomorphism preserves order of elements, which I proved couple of exercises back. I am also interested in other ways of proving it. Is there a more ...