Finding all Sylow 2-subgroups I want to find all Sylow 2-subgroups of $D_6 = \{ e, r, r^2, r^3, r^4, r^5, s, sr, sr^2, sr^3, sr^4, sr^5\}$.  $r$ is the rotation element, and $s$ is the reflection element.  $|D_6| = 12$, and the order of a Sylow 2-subgroup of $D_6$ must be the highest order of $2$ that divides $12$ which is $4$.  Also, the order of each element of the Sylow 2-subgroup must be a power of $2$.  Using this information I was able to come up with only this Sylow 2-subgroup $\{ e, s, r^3, sr^3 \}$.  However, I do not know if there are more Sylow 2-subgroups of $D_6$.  Lastly, I figured out, by Sylow's Third Theorem, that the number of Sylow 2-subgroups of $D_6$ must divide the order of $D_6$, and also be congruent to 1 mod 2.  The first part of this statement suggests that there can only be either $1,2,3,4,6$, or $12$ Sylow 2-subgorups of $D_6$, but I am unsure of what congruent to 1 mod 2 means.  I know that 1 mod 2 $= 1$, but what does congruent mean?
Finding Sylow 2-subgroups of the dihedral group $D_n$ I am trying to describe the Sylow $2$-subgroups of an arbitrary dihedral group $D_n$ of order $2n$.   In the case that $n$ is odd, $2$ is the highest power dividing $2n$, so that all Sylow $2$-subgroups have order $2$, and it is fairly easy to describe them.  However, if $n$ isn't odd, we may factor a power of $2$ out and write $|D_n|=2^{k}m$ for some odd integer $m$.   There is  that there exist precisely $m$ Sylow $2$-subgroups, but it does not provide an explicit description of such subgroups in the case $n$ is odd.   Additionally, someone has asked a similar question in the past, and they claim to give a description of the Sylow $2$-subroups in the case $n$ is odd, but I can not find a source or an explanation. The question is .     Can anyone provide a description of how one may determine precisely the Sylow $2$-subgroups for the case $n$ is odd?    Thank you.
Sorry if this is very basic but here's a question.   Let $\mathbf{v}=(v_1,\ldots, v_n)\in k^n$ where $k=\bar{k}$.      Why do we have   $$ \sqrt{\sum_{i=1}^n |v_i|^2} \leq \sum_{i=1}^n |v_i|,  $$   where the left-hand side can be thought of as the $2$-norm $\|\mathbf{v}\|_2$ on $L^2(k^n)$?   $\mathbf{General \; case}$: If this is true for $p=2$-norm, I am guessing that this is true for all $p\geq 1$:  $$ \left( \sum_{i=1}^n |v_i|^p\right)^{1/p}\leq \sum_{i=1}^n |v_i|.  $$     In fact, does this inequality hold when $p$ is a rational number? Text spirals with TikZ I just ran across this, and I'm curious about whether it can be replicated in TeX...    Ideally one would have the spiral getting tighter and tighter as one got to the centre, as in    I know TikZ has the ability to fit text along a path, but I'm not sure how one might make text get smaller and smaller.  The closest I can find is , but that uses manually inserted font commands.  Edit: I got what I wanted using mwibrow's answer: Is "there're" (similar to "there's") a correct contraction? Q: "Do you have any juice?"   A: "Yes, there's some in the fridge."   Sounds perfectly fine to me, but:     Q: "Do you have any towels?"   A: "Yes, there's some in the closet."   Does not.  I asked for towels - plural - so wouldn't "Yes, there're some in the closet," in which there are is turned into a contraction be the correct way to say it?  Spellcheck, however, doesn't like "there're", and I think I'm the only person I've ever heard use the word "there're".  Even folks who I know say "there are" shorten it to "there's" when possible.  Am I saying it wrong, or are both forms acceptable? Earth moves how much under my feet when I jump? If I'm standing at the equator, jump, and land 1 second later, the  Earth does NOT move 1000mph (or .28 miles per second) relative to me,  since my velocity while jumping is also 1000mph.   However, the Earth is moving in a circle (albeit a very large one),  while I, while jumping, am moving in a straight line.   How much do I move relative to my starting point because of this? I  realize it will be a miniscule amount, and not noticeable in practise,  but I'd be interested in the theoretical answer. On the Goblet of Fire Barty Crouch exactly got Mad-Eye's voice when he transformed into him, but on the Deathly Hallows, they were all able to have their own voices instead of Harry's. Why is the effect of the polyjuice potion inconsistent? Let $M$ be an A-module and $f \in \text{End}(M)$ an idempotent endomorphism. Show that $M = \ker(f) \oplus \ker (f-\text{id}_M)$. I don't know how to approach this exercise, using only properties of modules and direct sums, without projections. Can you help me? Thanks in advance. How do I set environment variables? I'm trying to set up Apache Tomcat on my pc, and it wants me to set up an environment variable for CATALINA_HOME.  Does any know how to do this? In RSA, why is it important to choose e so that it is coprime to φ(n)? When choosing the public exponent e, it is stressed that $e$ must be coprime to $\phi(n)$, i.e. $\gcd(\phi(n), e) = 1$.   I know that a common choice is to have $e = 3$ (which requires a good padding scheme) or $e=65537$, which is slower but safer.  I also know that for two primes $p,q$, we have $\phi(pq) = (p - 1) (q - 1)$  Now, let me give a (simple) example:  Say I choose $e = 3$, and two random primes $p = 5$ and $q = 13$.  I can now compute $\gcd(3, \phi(5 \cdot 13)) = 3$.  This reveals that $3$ and $\phi(n)$ are not coprime. I assume this could also happen for large values of $p$ and $q$, and likewise for another $e$. I therefore assume that the RSA algorithm must check that $\gcd(e, \phi(pq)) = 1$. But let's assume it doesn't.  How does RSA become vulnerable if $\gcd(e, \phi(pq)) \neq 1$? Why are some applications so hard to find in the Ubuntu Software Centre? Some applications don't pop up in the search even when you type in their name. Take for instance Sushi. When you type in "sushi" into the search box, it fails to throw up the application. Only when you type in "gnome-sushi" does it appear. This is despite the fact that the application has "sushi" by itself in the application description. There are other applications which don't spring to mind right now, but I've discovered before. How to export a individual layer with a fixed master layer in Photoshop? Let's say I have some layers and a single different one, or a group of layers I want to export merged together, in a way that every single exported layer is merged with a specific layer or group of layers.  In the example below I have some layers which need to be exported into JPG or PNG, merged with the very top layer called "Blurred". In my case, I want to export the layers (Jan 8 + Blurred) and (Layer 10 + Blurred) and so on. Acting as exporting layers merged with a master layer called "Blurred".  My question in a simple way: how to apply a Master layer(s)/Group(s) in Photoshop to be exported with every layer I have? Using "you" and "your" as a representation for yourself and everyone in general Example sentence:     I love when your dog just lets you sit there to pet them. You don’t necessarily know if they are enjoying it, but they love you enough to just sit there with you for a bit.   Is this correct? We assume the words "you" and "your" refer to the speaker of the sentence, and not to the listener, as second-person usually does. But it also refers to dog owners in general. I have always been curious about this. I used MiKTeX 2.9 on windows. I have downloaded a package I want to install from the internet (manually, not through MiKTeX). The archive (*.lmza) containing the package was saved to a directory on my C:\ drive. Through MikTeX -> Maintenance -> Settings -> Packages -> Change... I can select the directory in which I have saved the archive containing the package. However, if I choose Packages shall be installed from a directory and choose the location of the package, I get this error message: C:\my\package\is\here does not seem to be a local package repository. What does it take for a directory/folder to become a local package repository? Error: could not start the command in Texmaker when I run makeindex I'm trying to create a nomenclature but I'm getting the following error when I run makeindex:     Error: Could not start the command   This error appears even with this basic example:  \documentclass{article} \usepackage[refpage]{nomencl}  \makenomenclature  \begin{document} Example of nomenclature \nomenclature{A}{Area}  \printnomenclature \end{document}   In the Configure Texmaker editor I wrote: makeindex %.nlo -s nomencl.ist -o %.nls in the makeindex slot. Then I run:   PdfLaTeX MakeIndex   And the error appears... I also tried all suggestions , but nothing seems to work.  I have installed MacTex 2012 distribution, Mac OS X 10.7.5 A capacitor stores half the energy when charged from a battery each and every time? I understand that when a battery is connected to a capacitor that half the energy is stored in the capacitor while the other half is lost to heat or whatever other losses are involved once the capacitor is charged to the exact same voltage level of the battery.  Only a one time direct connection with a simple switch from open to closed that charges the capacitor up to the voltage level of the battery is what is being referred to. My question is does this occur each and every time?   Once the capacitor is charged to the voltage level of the battery then there should be 7.2 milliJoules of energy in the capacitor.  If the answer is yes that half the energy is transferred into the capacitor each and every time then what would the capacitor voltage have to be in order to equal 100% of the energy if it were to come from the battery?  I understand that in practice this will never happen, which is why the statement equal to 100% of the energy is indicated. (Meaning by some other method of charging means)         &ndash; Schematic created using What does the |= operator do in Java? Why is Linux's filesystem designed as a single directory tree? By default, all browser instances use the same cookie for the same site. Is there a browser that doesn't follow this?  Or is there a Firefox plugin that can "wear" different cookies for a specific site? Each time I open any application, it zooms to fill-up the entire desktop and so I am always forced to use "Unmaximize" using Alt+Space menu. GNOME 2 allows one to either save the session automatically or manually (i.e. save the session as it looks right now). This is a nice feature because it restores the various desktop states between logins (and system restarts):   which apps were open where (physically) on the desktop were they open what specific window sizes were these apps set to   Looking around, I can't find the feature in GNOME 3? Did I not look hard enough? Can you fake a closed question? [‌closed] I just had a random thought; what if you append "[closed]" to the end of a question title. Is it filtered out, or in some way disallowed?  If you are seeing this, it isn't. It could potentially be used for some trickery, right? Or maybe not. I don't know, that's why you're here: discuss!  EDIT: Just a recap: Juan Manuel edited and removed "[closed]" from the end of the title and then answered (see below) - but it is technically possible.  Edit 2: Looks like it's still possible, if you play silly tricks with Unicode. (There's a  character in the new title.)