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- Jan 26, 2012

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Thanks to those who participated in last week's POTW!! Here's this week's problem.

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Are $G$ and $H$ isomorphic under multiplication? If yes, prove it. If not, provide a counterexample.

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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**Problem**: Let $G=\{a+b\sqrt{2} : a,b\in\mathbb{Q}\}$ and let $\displaystyle H=\left\{\begin{bmatrix} a & 2b \\ b & a\end{bmatrix} : a,b\in\mathbb{Q}\right\}$ be two groups. Show that $G$ and $H$ are isomorphic as groups under addition; i.e. find a*bijective*map $\varphi:G\rightarrow H$ such that for any $x,y\in G$, $\varphi(x+y) = \varphi(x) + \varphi(y)$, where $\varphi(x),\varphi(y)\in H$.Are $G$ and $H$ isomorphic under multiplication? If yes, prove it. If not, provide a counterexample.

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Remember to read the POTW submission guidelines to find out how to submit your answers!

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