I know, $|S_4 |=24 =2^3\cdot3$. So here the order of sylow $2$ subgroup is $8$ and by the third sylow theorem we can say the number of sylow $2$ subgroups is $1$ or $3$. Then by finding the sylow $2$ subgroups explicitly we can conclude that its $3$. But here we also know the number of sylow $3$ subgroups is $1$ or $4$.  My question is can we find the exact number of sylow $2$ and $3$ subgroups without calculating the the sylow subgroups explicitly, just by element counting of the subgroups, because sometimes we need only the exact number in of sylow subgroups ?
I am trying to find all the Sylow 2 subgroups of S4 using Sylow’s theorems. Now, I know that a Sylow 2 subgroup of S4 has size 8, and that there are either 1 or 3 of them (as the number of of Sylow 2-subgroups has form 1+2k and divides 3, the index).  Now my lecturer states “stabilisers of the 3 different bisections of {1,2,3,4} yield 3 distinct Sylow 2 subgroups”  Now my questions are:  1) What does it mean by stabilisers of 3 different bijections of {1,2,3,4}? Is that to say the elements of S4 which send {1,2} and {3,4} to {1,2},{3,4} i.e. by (12)(34)?  2) How does he know that there are 8 elements of this set? Perhaps this will become clearer after the 1st question is answered.  Many thanks, group theory is hard.
How do websites block selecting text and how do I unblock that? There's a website () that somehow blocks selecting text. Also it blocks Ctrl+A to select everything, there is also no “Copy” in the popup menu.  What I have tried:  Some sites use JavaScript to block selection. So I disabled all JavaScript sources in no-script addon in Firefox. I tried the same site in Google Chrome with the Script Safe addon. Made sure that everything is blocked, still can’t select text. Mousepointer remains an arrow and not a text cursor also on the whole site.  Used about:config to disable JavaScript completely in Firefox. Still no effect.  I read that some sites use DIVs with the style display:block so I used Inspect to examine the styles of the site. There is not one mention of "block" on the whole website, not in embedded CSS nor in object style=-attributes.  The text is not an image or flash or some HTML5 canvas ect. Everything is within DIV and P tags but no style was found that might block the text selection.  How can the website still block any selection of text? Not to mention why browsers support such behaviours at all. I just want to use a selected word or sentence and want to search google for it using right mouse click. It’s pretty annoying when a website does such things and forces me to type technical terms and names into google by hand. It disturbs my workflow. Proving the Cauchy-Schwarz inequality by induction I ran across this problem in some old notes, and I frustratingly can't figure out how to do it   Let $a_i$ and $b_i$ be sequences of natural numbers, use induction to show  $\sum_{i=1}^n (a_ib_i)^{1/2} \le (\sum_{i=1}^n a_i)^{1/2}(\sum_{i=1}^n b_i)^{1/2} $  Obviously this is trivial to show for n=1. I can't make much progress on n+1. I've tried various tactics, squaring both sides etc.   Any hint or help would be appreciated. Thanks. I'm looking for a way to search files where two word instances exist in the same file.  I've been using the following to perform my searches up to this point:  find . -exec grep -l "FIND ME" {} \;   The problem I'm running into is that if there isn't exactly one space that between "FIND" and "ME", the search result does not yield the file.  How do I adapt the former search string where both words "FIND" and "ME exist in a file as opposed to "FIND ME"?  I'm using AIX. "...would have trouble imagining...": Is this present participle or gerund? Troubleshoot High CPU usage by the "System" process Is there a well-known term for the synonym or near-synonym "telescoping" words? This has been rattling around in the back of my mind for many years (way before Stack Exchange came into existence), so it's a relief to finally ask the question.  There are words that can be "contracted" by removing one or more letter(s) (without changing the order of the letters) to form a shorter word with the same meaning, or one very close to it.  The two strongest examples I can think of right now are:  rapscallion -> rascal  and  satiate -> sate  This is a slightly weaker example:  rapine -> rape  Even though "rape" has come, in modern usage, to refer almost exclusively to a sexual crime, it can still be used in a more literary (and purposefully anachronistic) sense to mean "the plunder and pillage of a country or region". In that sense, "rape" can be used in a similar fashion to "rapine"; in any case, the two words are etymologically related to the Latin rapere which is tied to the archaic usage.  Another weaker example:  transliterate -> translate  These words are near-synonyms, but there is a definite difference in their meaning. Still, I guess this pair would qualify, albeit in a looser sense.  I wanted to exclude some relatively trivial examples, such as:  (inflammable -> flammable)  (perambulate -> ambulate)  and so forth. So I've imposed the additional rule (arbitrary though it may seem) that the "stripped" letters must not come from the start or the end of the word, but only from its middle.  Acronyms and portmanteaus are also excluded, as are "slangier" contractions e.g.:  (modulator-demodulator -> modem)  (motor-hotel -> motel)  (certitude or certainty or certificate -> cert)  I guess the long-short forms of the same word would form a linguistic doublet. I've coined the term "telescoping word" to describe the long form that "collapses" into the short form with letter deletion sans anagramming.  My question is: has this phenomenon already been studied? If so, what name have authorities given it? Is there already an existing reference or resource on it? Thanks in advance.  (This is not a duplicate of , as that is asking about the more general phenomenon of words (not necessarily synonyms) occurring within larger words. In fact, the question specifically mentions "kangaroo word" and states that a broader scope is being sought.) Prove that there is no non-zero linear operator on $\mathbb C^{2}$ such that $(\alpha | T\alpha) = 0$ How should we handle a question which was deleted and reasked? A user has asked  which did not receive much attention, and thus the user deleted the post. After some time, the user asked the  (with just slight changes to the title and a tag change), so that it could receive a bit more attention. The strange thing is that the question was not downvoted or poorly written, and yet the user deleted the original post. Now, since the original question was deleted, I could not mark the new one as a duplicate (it threw an error). My question is, how do we handle such questions? Just let them be as such or let the user know about it or is there another way to handle such situations? I quite often add a question as a favourite when there is something of interest to me in it or its answers. Looking back through my favourites, I cannot remember why I added some of them as a favourite.  What I'd like to be able to do is to add a note/comment when I add a favourite as a reminder as to why I've added it. This will show on my favourites page, but nowhere else - and possibly to no one else.  In addition, it would be good to be able to favourite an answer to a question, thereby adding a favourite to both the question and a particular answer with something of note in it. This would only show as a single favourite, but when clicking its link would go to the favourited answer.  Possible? Or is there another way of doing this? I've already got too many favourites to remember why I added a lot of them. Solving the recurrence $T(n) = \sqrt n T(\sqrt{n}) + \sqrt{n}$ A former student of mine was TA-ing an algorithms class last quarter and asked students to solve this famous recurrence relation:  $$T(n) = \sqrt n T(\sqrt{n}) + n$$  There are  and this question has already been asked here.  I was talking to my student about this problem and mentioned that it's quite hard to solve and is pretty dependent on the particular choices being made. As an example, I suggested this variant of a recurrence relation that was less simple to solve:  $$T(n) = \sqrt n T(\sqrt{n}) + \sqrt{n}$$  The TA and I tried solving this recurrence for about an hour and a half without making any progress. Here are a few things we tried:   My approach to solving the initial recurrence relation was to draw out a recursion tree and notice that each level of the tree contributes $n$ to the total and that there are $O(\log \log n)$ layers, so the recurrence solves to $O(n \log \log n)$. When I tried doing this here, I noticed that the work per layer was no longer constant; instead, the top layer sums to $\sqrt{n}$, the second layer to $n^{3/4}$, the third to $n^{7/8}$, etc. We got stuck working with the summation $n^{1/2} + n^{3/4} + n^{7/8} + ...$. We tried using the iteration method to unroll the recurrence. With the original recurrence relation, this works out nicely; here, we got stuck at the same summation given above.   I'm completely stuck trying to figure out how to solve this recurrence relation. There's nothing riding on it per se - I don't need to solve it for any particular reason - but the fact that we arrived at it by a straightforward modification of a common algorithms problem set question makes it all the more enticing.  Any idea how to solve this recurrence?  Thanks! What is the best way to give a C# auto-property an initial value? Evaluating $\int\limits_0^{\pi} \frac{dx}{1+2\sin^2x}$ After making u substitution two times, I am getting indefinite integral as $$\int\limits\dfrac{dx}{1+2\sin^2x} = \dfrac{\arctan(\sqrt{3}\tan(x))}{\sqrt{3}}+ C$$  I am stuck at working the bounds because I am getting $0$ but the answer needs to be $\pi/\sqrt{3}$.  I think $\arctan(x)$ is defined in $-\pi/2 \lt x\lt\pi/2$ while making u substitution. However I don't see how to use this observation to my advantage. Can anybody help in getting the answer and understanding whats going on. Thanks ! When I apply a glossy shader to the material of my text, the text becomes just black. Why?    I have lamp and everything, but the problem persists. The text is red before I add the glossy shader.  I'm using Cycles render.   Thanks! Do you add Proficiency bonus AND the stat modifier to skill rolls? I know this basic knowledge, but I seemed to have missed this info in the player's manual.  Say someone makes an stealth check: do I add their Dexterity modifier and their proficiency bonus in Stealth to the roll? In other words, does the Dexterity modifier apply to all Dex-related skills, including ones with a proficiency? When A and B are positively related variables, can they have opposite effect on their outcome variable C? A is positively related to B.  C is the outcome of A and B, but the effect of A on C is negative and the effect of B on C is positive.  Can this happen? Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$ Migrated question's answers can be upvoted twice by the same user was created on Server Fault, answered on Server Fault, I upvoted some of those answers on Server Fault, and then moved by consensus to Super User.  Now that the question is on Super User, I can upvote the same answers I previously upvoted on Server Fault. One-sided limits of $f'(x)$ at a point vs. one-sided limits of the difference quotient at that point I know that if a function is piecewise, then when differentiating it one needs to deal separately with the points of intersection of the pieces, because the derivative may not be defined there.  Also, if a one-sided limit of $f'(x)$ at these "crucial points" is infinite or doesn't exist, it doesn't preclude the limit of the difference quotient from existing and being finite. For example, take $f(x)=x^2\sin\frac1x, f(0)=0$: $f'(x)= 2x\sin\frac1x-\sin\frac1x $ has no limit as $x\to0$ but $\lim\limits_{h\to0}\frac{f(h)-f(0)}{h}=0=f'(0)$.  On the contrary, looking at $f(x)=\sin x +\lvert \sin x\rvert $ on $[0,2\pi]$ one has $f'(x)=2\cos x$ on $(0,\pi) $ and $f'(x)=0$ on $(\pi,2\pi)$ so $\lim\limits_{h\to0^+}f'(\pi+h)\ne\lim\limits_{h\to0^-}f'(\pi+h)$, but they are both finite. And in this case it's easy to find that they agree with the one-sided limits of the difference quotient. Why? Does the finiteness of the former imply that they agree with the latter? Is this also true if the one-sided limits of $f'$ at the crucial points agree with each other? I install virtualBox on Ubuntu 16.04, but after install and create new virtual Host shows this error:     Kernel driver not installed (rc=-1908)      The VirtualBox Linux kernel driver (vboxdrv) is either not loaded or there is a permission problem with /dev/vboxdrv. Please install virtualbox-dkms package and load the kernel module by executing  modprobe vboxdrv       as root. If it is available in your distribution, you should install the     DKMS package first. This package keeps track of Linux kernel changes and   recompiles the vboxdrv kernel module if necessary.  where: suplibOsInit what: 3 VERR_VM_DRIVER_NOT_INSTALLED (-1908) - The support driver is not installed. On linux, open returned ENOENT.     How to address this modprobe vboxdrv error? font-not-found error: font "HoeflerText" cannot be found -- LuaLaTex error! I am running this CV/Resume example from :   Here is a relevant fraction of the code:  \documentclass[10pt]{article} %!TEX encoding =  UTF-16 % See geometry.pdf to learn the layout options. There are lots. \usepackage[hmargin=1.25cm, vmargin=1.5cm]{geometry} %symbols - the ones you see on the left of the email and of the phone \usepackage{bbding} \usepackage{amssymb} \usepackage{eurosym} %Colors/Graphics \usepackage{color,graphicx} \usepackage[usenames,dvipsnames]{xcolor}  %Fonts and Tweaks for XeLaTeX \usepackage{fontspec,xltxtra,xunicode} \defaultfontfeatures{Mapping=tex-text} \setromanfont[Mapping=tex-text]{Hoefler Text} ...   So... when compiling with LuaLaTeX (part of MiKTeX on Windwos XP x64), I am getting the following error:  ... luaotfload | Font names database loaded: C:/Documents and Settings/Ampleforth/L ocal Settings/Application Data/MiKTeX/2.9/luatex-cache/generic/names/otfl-names .lua luaotfload | Updating the font names database: luaotfload | Scanning TEXMF and OS fonts... luaotfload | Font names database saved: %s  : C:/Documents and Settings/Ampleforth/Local Settings/Application Data/MiKTeX/2 .9/luatex-cache/generic/names/otfl-names.lua  !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !  ! fontspec error: "font-not-found" !  ! The font "HoeflerText" cannot be found. !  ! See the fontspec documentation for further information. ! For immediate help type H &lt;return&gt;. !...............................................    l.28 \setromanfont[Mapping=tex-text]{Hoefler Text}   What is wrong? How can I install this font so LuaLaTeX is picking it up? Where to download? Any other thoughts about this?