Does there always exists an injective homomorphism from $G$ into $S_m$ for some $m&lt;n$ where $|G|=n$   I know that for $|G|=n$ there is always an injective homomorphism from $G$ into $S_n$
Let $G$ be a finite group of order $n\ge2$. Which of the following statements are true? a. There always exists an injective homomorphism from $G$ into $S_n$. b. There always exists an injective homomorphism from $G$ into $S_m$ for some $m\lt n$. c. There always exists an injective homomorphism from $G$ into $GL_n(R)$.  I think b. is right (don't know how to prove though). I also think that c. is false. But not sure.
I'm trying to identify a science fiction series on UK TV during the 70s, but I can only recall the following:   A party of adults and children. There was a woman with a telepathic bond to her cat: she could look into its eyes to see what it had seen. Time travel was involved. I think it may have been inadvertent or uncontrolled time travel. On one occasion the party encountered an Elizabethan seaman who demanded to know if Elizabeth was still queen. They party replied "err... yes" because they didn't want to have to explain time travel. What is the translation of what's written below?  I think it may be the poem of the One Ring, but I'm not sure. I tried to install ffmpeg in trusty/Ubuntu 14.04 and got the following message:  $sudo apt-get install ffmpeg  Reading package lists... Done Building dependency tree        Reading state information... Done Package ffmpeg is not available, but is referred to by another package. This may mean that the package is missing, has been obsoleted, or is only available from another source  E: Package 'ffmpeg' has no installation candidate   Why isn't the package available? How to add existing user to an existing group? I want to add the Apache user (www-data) to the audio group. I've read the man page for useradd, but I'm not having any luck. I'm running xubuntu 11.10. Here's what I'm doing:  $ sudo useradd -G audio www-data useradd: user 'www-data' already exists   If I leave out the -G option, bash, prints the help info for useradd:  $ sudo useradd  audio www-data Usage: useradd [options] LOGIN Options: -b, --base-dir BASE_DIR       base directory for the home directory...   It's not clear to me from the man page what options I should use to make this work. Standard packages for compact SMT crystals There is an old and bog standard form factor for SMT crystals - HC49/US. The problem is that HC49/US takes a lot of real estate on the board.  It's 12mm long.  Another problem with crystals and oscillators in general is that they often become long lead time items.  [This is my experience.]  There are lots of small form factor crystals from various manufacturers.  But, at a glance, it seems that there is no standardization of form factor.  It’s nice when packages are standardized.  It alleviates the long lead time problem.  Crystals from manufacturer B may be used, if crystals from manufacturer A are out of stock.  Are there standard form factors for compact SMT crystals?  Update  Armed with the insights from Gribo and Gustavo, I did a more focused survey.  Found that 12MHz in 3.2x2.5mm package are produced by: Abracon, Citizen, Epson, Fox, ECS, TCS, NDK.  To be included in this list, the part had to be in stock at DigiKey.  It turns out that 3.2x2.5mm package is common across many manufacturers. How do I install OpenStack? Your last flag was declined - no it wasn't 3 more votes from other users are needed to close this question.   This is the message I got hovering the mouse over close voting tab.   I wonder can we begin any sentence with a numeric value? Is it good practice? One thing I liked about Mint was that the terminal would usually show two colors: everything from the left to the $ would be one color, and everything past the $ would be another.   Is there a way to have this setup with Ubuntu? A command-line clipboard copy and paste utility? In Windows I used command-line clipboard copy-and-paste utilities...      pclip.exe and gclip.exe     These were UnixUtils ports for Windows (but they only handled plain text). There were a couple of other native Windows utilities which could write/extract any format.    I've looked for something similar in Synaptic Package Manager, but I can't find anything.    Is there something there, that I've missed? ... or maybe this is available in Bash scripting?    The type of utility I'd like will be able to read/write via std-in/std-out or file-in/file-out, and handle Unicode, Rich Text Format, picture, etc. clipboard formats...    NB: I'm not after a clipboard manager. What does "rc" in .bashrc stand for? Is it "resource configuration", by any chance? Which is easier, pushing or pulling? It is generally assumed, from a person's perspective, that pushing a cart is more easier than pulling one. But why? Is there any difference in terms of force required to achieve the same amount of displacement?  Or is it merely human perception?  Why is it that almost all automobiles transfer torque to the back axle. But then, why do trains have engines in the front? Spacing issue for newenvironment wrapper of equation environment If I create a custom environment based on the equation environment, I get a spacing issue: the next line of text after my environment closes doesn't start on the left margin.  Example code:  \documentclass{article} \begin{document} \newenvironment{myenv}{\begin{equation}}{\end{equation}} \noindent $| \leftarrow$ this is the left margin. \begin{myenv} e ^{i \pi} + 1 = 0 \end{myenv} $| \leftarrow$ this isn't the left margin. \end{document}   Output:    Removing the newline after \end{myenv} solves the spacing problem, but the resulting code is not readable.  Of course, in this example I could just use the equation environment itself. My real issue involves a more complex custom environment that suffers from the same problem, and which also includes an equation environment within it. Order of products of elements in a finite Abelian group We want to show that if $a,b\in G$ where $G$ is a finite Abelian group, we have  $\operatorname{LCM}(|a|,|b|) = |ab|$ given that $ab \neq e$.   How I approached this question was by saying let $\operatorname{LCM}(|a|,|b|) = L$. Then if we show that $L$ divides $|ab|$ and $|ab|$ divides $L$ then $|ab| = L$. Showing that $|ab|$ divides $L$ was fine. But then I am having trouble with the second part that is showing $L$ divides $|ab|$. For this, so far I have:  Consider $(ab)^{|ab|} = e = a^{|ab|}b^{|ab|} $  since we know $a \neq b^{-1}$ by assumption then we can conclude that $|a|$ divides $|ab|$ and $|b|$ divides $|ab|$. We know that $L = \frac{|a||b|}{\operatorname{gcd}(|a|,|b|)}$ from here can we conclude that $L$ divides $|ab|$? Both myself and my boyfriend have PS4s.  Can I just get a single PSN account and sign both consoles into the same account, or do I need to get two separate accounts? Django: How can I protect against concurrent modification of database entries Why the following example prints "0" and what must change for it to print "1" as I expected ?  #include &lt;iostream&gt; struct base {    virtual const int value() const {       return 0;    }    base() {       std::cout &lt;&lt; value() &lt;&lt; std::endl;    }    virtual ~base() {} };  struct derived : public base {    virtual const int value() const {       return 1;    } };  int main(void) {    derived example; } Here's a module with castellated pads attached to an eval board:  (source: ) Looking closely at the wires interfacing with the castellated pads, they're clearly soldered to the eval board. However, they don't look like they're soldered to the module, but rather they're being pressed against the pads with the tension in the wire. It also looks like the free end of the loops are protruding down into a channel cut in the PCB. Do these kind of connectors/metal loops have a name? Why does cold metal seem colder than cold air? (I apologize for this elementary question. I don't know much about physics.)  Let's say that I put a metal pot in the refrigerator for several hours.  At this point, I guess, the pot and the air (in the refrigerator) have the same temperature.  Now, I touch this pot. It feels very cold. But when I "touch" the air (that is inside the fridge) it doesn't "feel" as cold. I don't feel the same "ouch!" that I feel when I touch the pot.  Why is that? Why does the metal seem colder than air although they both have the same temperature?  (I know that gas has less particles in it in one unit of volume compared to solids and liquids, but since "temperature" means "the average kinetic energy", these fewer air particles are supposed to hit my hand in a velocity that's going to compensate for their lower number, aren't they?)  A related question, for clarification:  If I use a thermometer to measure the temperature of the pot &amp; air (let's assume it's a thermometer that has a probe that can touch objects), will it show the same reading for both? If so, what makes the thermometer different than my hand? I mean, my hand is sort of a thermometer, so why would it fail whereas a non-human thermometer would work? Java : How to determine the correct charset encoding of a stream