Automorphisms of a direct product of cyclic groups
Number of automorphisms of a direct product of two cyclic $p$-groups
Zombies - Why do they need to feed on living animals/humans? Any command that starts with $ returns  command not found   How can I fix this?  I am  using 16.10 Ubuntu. Can I run OSX applications? Show the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$ Show that for $n \geq 5$, the commutator subgroup of $S_{n}$ is $A_{n}$ for $n \geq 5$.     I'm not sure on how to handle this problem. I know that the commutator subgroup $G'$ will consist of products of commutators $[\sigma_{i}, \tau_{i}]$ for permutations $\sigma_{i}, \tau_{i} \in S_{n}$. However, I don't know where to start. What would be the best way to approach this? Archive by Year How can you encode a string to Base64 in JavaScript? I have a PHP script that can encode a PNG image to a Base64 string.  I'd like to do the same thing using JavaScript. I know how to open files, but I'm not sure how to do the encoding. I'm not used to working with binary data. Reference for creative set criterion theorem Proving $M_n(I)\lhd M_n(R)$ and proofs about ideal rings. I'm trying to do a few proofs about rings: $(1)$ I want to show that for a ring $R$ and the ring of matrices $M_n(R)$, if $I\lhd R\implies M_n(I)\lhd M_n(R)$ and that all ideals of $M_n(R)$ are of the form $M_n(I)$. $(2)$ Prove that if $R$ is a ring with division then $M_n(R)$ is a simple ring.  The relation $I\lhd R$ implies that with $r\in R$ and $x\in I$ we have $rx\in I, xr\in I$. But how can I prove the statement for matrices?, here's what I would do: Consider $\psi \in M_n(R)$, if is given any matrix $\phi\in M_n(I)$ it must be $\psi \phi \in M_n(I),\phi \psi \in M_n(I)$, if we denote the element $(i,j)$ of each matrix by $\phi_{ij}$ and $\psi_{ij}$, then $(\psi \phi)_{ij}=\sum_{k=1}^n \psi_{ik}\phi_{kj}$ and $(\phi \psi)_{ij}=\sum_{k=1}^n \phi_{ik}\psi_{kj}$.  Since $I\lhd R$ is $\psi_{ik}\phi_{kj}\in I$ and $\phi_{ik}\psi_{kj}\in I$, which means that $I\lhd R\implies M_n(I)\lhd M_n(R)$.  To show that every ideal of $M_n(R)$ are of the form $M_n(I)$ with $I$ ideal: Suppose $I$ is not an ideal of $R$, then exists $r\in R$ such that $rx\notin I$ or $xr\notin I$, then with taking a matrix $\phi\in M_n(I)$ such that $x$ is an entry of $\phi$ and a matrix $\psi$ such that $r$ is an entry of $\psi$ we get that the matrices $\phi$ and $\psi$ have an entry $xr$ or $rx$, in either case this is not an element of $I$ then the matrix is not in $M_n(I)$. This concludes my proof (is it?) for $(1)$.  For $(2)$ I'd like to proceed by absurdum . Let's say that $M_n(R)$ is not a simple ring, then it has at least one ideal. Denote by $\varphi$ this ideal in $I$, then for any matrix $\psi$ we have $\varphi\psi \in M_n(I)$ and $\psi \varphi \in M_n(I)$.  Now, can $R$ be a ring with division?.I'm not sure how to prove that it can't. I believe I could use this theorem: "If $I\lhd R$ and $1\in I$ then $I=R$. The case $I\neq R$ implies $1\neq R$ and since $\varphi \psi$ and $\psi \varphi$ are in $M_n(I)$ then there are at entries in the product matrix which belong to $I$ (using $I \lhd R$) but are not $1$ (1\notin I); then there are elements in $I$ (and therefore in $R$ by $I\subset R$) that are not invertible, follows that $R$ is not a ring with division. But if $I=R$?, here I can't see a why $R$ can't be a ring with division. Setting up LVM Snapshot as a backup/restore point in ubuntu. Why? Well, as I see it, with all the great usefulness of Linux, its biggest drawback is with its mixed up way of installing apps. All the dlls go to one directory. All the configuration files to another. And the binaries are also put in a bunch. Yes, Windows works the same way more or less, but with the nature of system/server software installed on Linux, being that most of the software installed is only about 95% mature at any given time, messing up your system twice a month is very probable.. Of course, we can backup the system before and after each installation, but that takes a million years. System backups (as oppose to just data/files backup) requires you to turn off you computer and use some form of bootable live CD and a backup media. And, if the drive is more than 15% full, it might take a long time to do the backup, even with modern hardware. So, how can this be done? Good tool for finding photo duplicates based on image data What is a good (and preferably free) tool for finding photo duplicates based on the photos themselves?  So, not filename, filesize etc., but how the photos look like. It would be nice if it managed to match photos that were the same image, but not exactly the same size for example too. How can I prevent my parents' divorce? I am a young teenager, younger than 16. (I do not want to give my age online)    My dad told me he wants a divorce, but I don't want my parents divorced. My dad says he can't stand my mom anymore.  My dad is older than 45 and my mom is older than 40.    If it happens, I would choose to stay with my mom. My dad can cook and drive, but my mom can't. There's a school bus in high school though.    What can I do to prevent this divorce? Analyze a circuit with two diodes and two voltage rails I want to find \$I\$, \$I_{D2}\$ and \$V\$ in this circuit. The diodes are made of Silicium. I'm a beginner and normally I would assume the Diode bias and then apply KVL to see if it is true. However, I've only encountered circuits where the wire forms a loop.    How am I supposed to apply KVL in this case? If voltage is the difference in potential between two points, how can I find the voltage of V if one point is outside of the circuit, therefore, outside of KVL application? Order of a product of elements in a generic group I'm dealing with the following question: having a group $G$ (of which nothing else is specified, doesn't have to be abelian), prove that $|ab|=|ba|$ for any $a$, $b$. Since $G$ is not abelian I'm not sure what to use here. I was thinking that except the case where $b=a^{-1}$, the order of those would be $lcm(m,n)$ where $m$ and $n$ are the orders of a and b. But looking it up it seems not to be so simple, even if all the info I could find was for abelian groups, so I would like some help with this specific case. Problem working fontspec, babel and datetime with cyrillic I have discovered that fontspec with russian babel settings doesn't work with datetime package. I tried to compile such code:  \documentclass[a4paper,11pt]{article} \usepackage[T2A]{fontenc} \usepackage[utf8]{inputenc} \usepackage[main=russian,english]{babel} \usepackage{datetime} \usepackage[cm-default]{fontspec}  \begin{document} \today \end{document}   with a xelatex and got an error: Command \cyri unavailable in encoding EU1.  Is there any way to handle it?  UPDATE: I really bungle with charset in XeLateX. Thanks to egreg. I have removed fontenc and inputenc and used some cyrillic font, so my MWE now looks this way:  \documentclass[a4paper,11pt]{article} \usepackage[main=russian,english]{babel} \usepackage{fontspec} \setromanfont{PT Sans} \usepackage{datetime}  \begin{document} \today \end{document}   and I've got another error, if I used russian as a main language in babel: Undefined control sequence. \\monthnamerussian ...cyrm \cyra \cyrya \or \cyri  \cyryu \cyrn \cyrya \or \c...  Is it still because of XeLaTeX? Proving that closed (and open) balls are convex Does the alarm work when my phone is turned off? Show that $3p^2=q^2$ implies $3|p$ and $3|q$ How do you reset timers? What does it mean for a QFT to not be well-defined? It is usually said that QED, for instance, is not a well-defined QFT. It has to be embedded or completed in order to make it consistent. Most of these arguments amount to using the renormalization group to extrapolate the coupling up to a huge scale, where it formally becomes infinite.  First of all, it seems bizarre to me to use the renormalization group to increase resolution (go to higher energies). The smearing in the RG procedure is irreversible, and the RG doesn't know about the non-universal parts of the theory which should be important at higher energies.   I have read in a note by Weinberg that Kallen was able to show that QED was sick using the spectral representation of the propagator. Essentially, he evaluated a small part of the spectral density and showed that it violated the inequalities for the field strength $Z$ that are imposed by unitarity + Lorentz invariance, but I do not have the reference.  So what I would like to know is :   What does it take to actually establish that a QFT is not well-defined?   In other words, if I write down some arbitrary Lagrangian, what test should I do to determine if the theory actually exists. Are asymptotically free theories the only well-defined ones?   My initial thought was: We would like to obtain QED (or some other ill-defined theory) using the RG on a deformation of some UV CFT. Perhaps there is no CFT with the correct operator content so that we obtain QED from the RG flow. However, I don't really know how to make that precise.   Any help is appreciated. Remove a part of a list of names I have some files and each files have distinct name but all of them have the same "-statement" at the end like this:  flower-statement.mp3 leave-statement.mp3  ball-statement.mp3 goal-statement.mp3   So I want to remove this: -statement  Is that possible?