A group $G$ is called perfect if $G$ coincides with its commutator subgroup, which is generated by the elements of the form $xyx^{-1}y^{-1}$, with $x,y\in G$. Let $SL_2(\mathbb{Z})$ denote the group of $2\times 2$ matrices with integer entries and determinant $1$. I think it is not perfect, but I do not know how to prove it, maybe by giving a counter example.
So far, I know the matrices $$S=\left( \begin{matrix} 0&amp;-1\\ 1&amp;0 \end{matrix} \right), \quad T=\left( \begin{matrix} 1&amp;1\\ 0&amp;1 \end{matrix} \right) $$ generate SL$(2,\mathbb{Z})$. $S^2=(ST)^3=-I$. I also know the matrices $$X=\left( \begin{matrix} 1&amp;1\\ 1&amp;2 \end{matrix} \right), \quad Y=\left( \begin{matrix} 2&amp;1\\ 1&amp;1 \end{matrix} \right) $$ with their inverses $X^{-1},Y^{-1}$ generate SL$(2,\mathbb{Z})'$, the commutator subgroup. I'm interested in what the quotient $SL(2,\mathbb{Z})/SL(2,\mathbb{Z})'$ is. In general, how would I go about computing the quotient of two matrix groups? I'm trying to use GAP but I'm having a lot of trouble!
Auto-mounting network shares per user I have a server that has a number of CIFS shares that need to be accessed on a per-user basis. For example, I have a Music share which I have full access to, but my wife has read-only access.  When either myself or my wife log into our Ubuntu 11.04 laptop I would like these shares to be automatically mounted per user. Now I understand that if I mount as -t cifs without specifying a user then it will use the USER environment variable. However, I also need to specify a password, so how can I do that when each user has a different password?  I think my questions are:   Is there a way for me to have a per-user /etc/fstab? If not, is there a way to specify that a mount is only applicable to a certain user? Also, the share password is always the same as the local password. Is there a way to specify that this password should just pass through from the client to server rather than having to specify it in a credentials file somewhere?   Or maybe I'm missing something and there's a completely different solution. Can anyone help? Why is apt-get always using proxy, although no proxy is configured? How can the pixel values of a raster image be extracted under a polygon fishnet? I have a single band raster image of 30m by 30m resolution. I have already created a polygon fishnet (grid) of 300m by 300m with point labels over that. How can I extract the mean pixel value of the raster image overlaying each feature in the fishnet grid? Is it possible with Arcgis?  Please inform me. How to hyperlink only the year part when using natbib and hyperref I am using the natbib and hyperref packages:  \usepackage{natbib}  \usepackage[pagebackref=true, plainpages=false, bookmarks, bookmarksnumbered, colorlinks, linktocpage=true, linkcolor=blue, citecolor=blue, filecolor=black, urlcolor=blue]{hyperref}   With my current settings (above) both the name and the year are hyperlinked.  My question is, whether there is a way to hyperlink (or color) only the year-not the author(s).   I have the impression reading this post  that biblatex does it by default. However changing to biblatex is not an option for me at this point.  Any helpful comments will be greatly appreciated. Can anyone please tell me whether the following function $d:\mathbb{R}^2 \times \mathbb{R}^2 \to  \mathbb{R}$ is really a metric or not on $\mathbb{R}^2$.  $$ d((x_1,x_2),(y_1,y_2))= \begin{cases} |x_1-y_1|, &amp;\text{ if } x_2=y_2\\ |x_1|+|y_1|+|x_2-y_2|, &amp;\text{ if } x_2 \neq y_2. \end{cases} $$  If it really is indeed a metric, please give the proof of triangle inequality.(I'm OK with the other two conditions). One of the new features in SQL Server 2012 codenamed Denali is the   I know a good bit about regular old row-store indexes, like the b-tree structure, differences in storage between the leaf level and b-tree pages, affects of included fields, optimizing to use them, order of keys, etc.  I'm having difficulty getting any good info on the internals of a columnstore index.     How is it structured? Is there a b-tree?  Some other structure in place? How is data organized? What sorts of specific operators are best suited to use it? Any other anti-patterns to be avoided when using them?   A lot of what I can find out about them are basically the exact opposite of a "normal" index, i.e. no ordering of keys, no included fields, nonclustered ONLY.  Any insights are appreciated. Make Launchpad Appear on screen with mouse Beamer Bibliography Icon I want to list my references in a beamer presentation using thebibliography environment, with the nice little    book icon (\beamertemplatebookbibitems) and  article icon (\beamertemplatearticlebibitems)   in front of each source.  Is there an icon for websites too? something like \beamertemplatewebsitebibitems? If not, is there a way to implement this easily ? How to use iSight webcam without activating the green light? Recently a the Dutch government advised to put a tape on your webcam to prevent hackers from taking pictures and video using your device ().  The build in iSight camera in Apple laptops have a green light next to the camera, this light turns green when you use the camera.  Is it possible to use the camera without the activation of the green light? Personally I think the green light turns on when the sensor requires power and this is hardwired in the hardware. Am I wrong?  I have found similar questions on stackexchange, but I found non that answer this specific question. Is there a limit to a total number of months a holder of UK multiple entry visitor visa can spend in UK per calendar year? How to ‘guess’ if a noun is countable or uncountable? "Please do not repeat the mistake" and "Please do not repeat the mistake again" sequence $x_{n+1}=x_n+e^{-2018x_n}$ Proving that two matrices are congruent, in order to establish a rank and signature result. In expressions such as "ham 'n eggs", the conjunction 'n appears to replace and, yet there is only one apostrophe to indicate the missing a and none for the missing d (i.e., no "ham 'n' eggs").    Is there a punctuation rule that governs this or is it just an idiosyncratic case? Uniqueness in existence of a bilinear form Let $E$ be a finite dimensional vector space over field $\mathbb R$ with $E^*$ as the dual space of $E$.($\dim E=n$)   $\Omega^P(E):=\{ \alpha\colon \overbrace{E\times\cdots\times E}^{p- times}\rightarrow \mathbb R\  \ , \alpha \text{ is alternating multilinear map}\}$ $\Omega^P(E^*):=\{ u\colon \overbrace{E^*\times\cdots\times E^*}^{p- times}\rightarrow \mathbb R\  \ , u \text{ is alternating multilinear map}\}$  I wish to prove that there exist an $\textbf{unique}$ bilinear form $B\colon \Omega^p(E^*)\times\Omega^p(E)\rightarrow\mathbb R$   such that for all $ f_i\in E^* \ , u_j\in E^{**}\simeq E \qquad  i,j=1,\cdots,p$, it has the following rule:  $B(u_1\wedge\cdots\wedge u_p,f_1\wedge\cdots\wedge f_p)=\det[f_i(u_j)]$.  $\wedge$ is wedge product between alternating maps i.e. $\alpha\wedge\beta(v_1,\cdots,v_{p+p})=\dfrac{1}{p! p!}\displaystyle\sum_{\sigma\in S_{p+p}}sign(\sigma)\alpha(v_{\sigma(1)},\cdots,v_{\sigma(p)})\beta(v_{\sigma(p+1)},\cdots,v_{\sigma(p+p)})$  How can I do this? How to subtract days from a plain Date? How can I really disable the Cmd to Win key mapping in Parallels Desktop 5? Removed it from System Preferences &gt; Keyboard. No go. Remapped ⌘ Cmd to be Ctrl. No go. Ripping out the ⌘ Cmd key shouldn't be the solution. Anyone?... Zero-dimensional ideals and finite-dimensional algebras I encountered in the literature the term zero-dimensional ideal, however I can not find a relevant definition anywhere in Atiyah-MacDonald or Matsumura. In fact, I encounted the statement:      $I$ is a zero-dimensional ideal of $k[x_1,\dots,x_n]$ $\Leftrightarrow$ $k[x_1,\dots,x_n]/I$ is a finite-dimensional $k$-vector space, where we assume that $k$ is algebraically closed.   Question 1: What is the definition of the dimension of an ideal?  Question 2: How do we prove the above statement?  Question 3: Is $k$ being algebraically closed a necessary condition for the statement to be true? Why is the gate charge curve (Miller plateau) of MOSFETs dependent on Vds?