Intereting Posts

Given $a\in\mathbb{R}^2\backslash X$ and $v\in\mathbb{R}^2$, $\exists\delta$ such that $t\in[0,\delta) \Rightarrow a+tv\in \mathbb{R}^2\backslash X$.
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Find the number of non-zero squares in the field $Zp$
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Prove $\left(\frac{e^{\pi}+1}{e^{\pi}-1}\cdot\frac{e^{3\pi}+1}{e^{3\pi}-1}\cdot\frac{e^{5\pi}+1}{e^{5\pi}-1}\cdots\right)^8=2$
cannot be the value of the expression.
Why is $e^x=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n$
Prove $\int_0^1\frac{\ln2-\ln\left(1+x^2\right)}{1-x}\,dx=\frac{5\pi^2}{48}-\frac{\ln^22}{4}$
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Prove that $\mathbb{N}$ with cofinite topology is not path-connected space.
Prove that for any infinite poset there is an infinite subset which is either linearly ordered or antichain.
A necessary condition for a multi-complex-variable holomorphic function.
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Why does topology rarely come up outside of topology?

Let $G\times K$ be a finite group. Suppoose $G\times K\cong H\times K$. Is this sufficient to imply $G\cong H$?

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Yes. In fact you only need to assume that $K$ is finite; that is, finite groups are cancellable. This is a theorem due to Hirshon.

Vipul Naik found an elementary proof:

For any finite groups $L$ and $G$, let $h(L,G)$ denote the number of homomorphisms from $L$ to $G$ and $i(L,G)$ denote the number of monomorphisms from $L$ to $G$. Notice that $$h(L,G)= \sum\limits_{N \lhd L} i(L/N,G) \hspace{1cm} (1)$$

Let $G,H,K$ be three finite groups such that $G \times H \simeq G \times K$. Then \begin{gather}h(L,G \times H)=h(L,G \times K) \\ h(L,G)h(L,H)=h(L,G)h(L,K) \\ h(L,H)=h(L,K)\end{gather} for any finite group $L$, since $h(L,G) \neq 0$. Using $(1)$, it is easy to deduce that $i(L,H)=i(L,K)$ for any finite group $L$ by induction on the cardinality of $L$. Hence $$i(H,K)=i(H,H) \neq 0.$$

Therefore, there exists a monomorphism from $H$ to $K$. Since $H$ and $K$ have the same cardinality, we deduce that $H$ and $K$ are in fact isomorphic.

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