I am working on the problem:     If G is a group of order 2n, show that the number of elements of G of order 2 is odd.   which already has solution (in the link below), please explain the solution in that link
I am working on the following problem from group theory:     If $G$ is a group of order $2n$, show that the number of elements of $G$ of order $2$ is odd.   That is, for some integer $k$, there are $2k+1$ elements $a$ such that $a \in G,\;\; a*a = e$, where $e$ is the identity element of $G$.
Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$ Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$.    Challenge: Prove  $$\sum_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}.$$   I have two proofs, one of which is partially combinatorial.    I'm posing this problem partly because I think some folks on this site would be interested in working on it and partly because I would like to see a purely combinatorial proof.  (But please post any proofs; I would be interested in noncombinatorial ones, too.  I've learned a lot on this site by reading alternative proofs of results I already know.)  I'll wait a few days to give others a chance to respond before posting my proofs.  EDIT: The two proofs in full are now given among the answers. I am trying to draw a partial box around one or more words within a paragraph--that is, a rectangle with one or more sides missing. I have read the documentation for the fancybox package and it doesn't seem to be able to do what I want. I was able to produce thus ugly hack:    That's just a vertical bar | followed by a word with \overline and \uline. The trick is to connect the vertical and horizontal lines so that you have a rectangle missing just one side.  Is there an easy, general way to do this, perhaps with Tikz? I don't know enough about that package to be able to do this myself, but I'm wondering if there might be other solutions too. When the loop is the figure shown below will be rotated in clockwise direction the the slip ring connected to the red segment of the loop will give us the positive voltage and the slip ring connected to the black segment of the loop will give us negative voltage according to the Flemings right hand rule.  What if the loop is rotated in counter clockwise direction ? Will the slip rings provide us the same polarities which they were giving when the loop was moving clockwise ?  Plus  The voltage from the loop is given by the equation.  V(t) = Vmaximum * sin (theta)  Where theta is the positive angle the red segment makes in the clockwise direction or in counter clockwise direction from this position ? Number of normal subgroups of a non abelian group of order $21$... CSIR December $2013$ Question is to :  Find the number of Normal subgroups of a nonabelian group $G$ of order $21$ other than $\{e\}$ and $G$.  What I have done so far is :  As $|G|=21=3\cdot7$  we have :  No. of sylow $3$ subgroups $1+3k$ dividing $7$ leaving out possibilities $1$ or $7$  No. of sylow $7$ subgroups $1+7k$ dividing $3$ leaving out only possibilities $1$.  So, we have a unique sylow $7$ subgroup and so it is normal.  I remember somehow that any normal group should come from a normal sylow subgroup or something like that.  S0, I prefer to conclude there is only one Normal subgroup for a non abelian group of order $21$.  Please let me know if this is true and please help me to fill that gap :      I remember somehow that any normal group should come from a normal sylow subgroup or something like that.   Thank you. I am upgrading my PC, but I'm sticking to my old HDD and gfx card. I would like to avoid reinstalling Win 8. Is this possible? Or, at least, is it possible to use Windows 8's refresh pc option?  UPDATE  It worked like a charm. Windows acted as if it was the first boot and displayed the "Getting devices ready %  " progress bar. I am not sure if this proves you can do it in every scenario, but my new gear is relatively modern (some gigabyte board with two PCIe 3.0 slots and USB 3.0 support + i5), so I guess we can agree the reinstall is unnecessary in most cases. Tikzpicture-figure: Order plot data (asc/desc)? I am using a tikzpicture for plotting. the values I want to plot are typical (X,Y) pairs. The x-value is an uninteresting id-value. I want to order the bins ascending/descending according to the y-value, but how do I do that? The figure seems to automatically take the x-values for ordering, how can I override that? (I am not particularly fixed to tikzpicture - its just the first thing that actually worked and displayed the plot....) Following minimal-working example, how could I achieve a reordering according to the y-value?  \documentclass{article} \usepackage[english]{babel} \usepackage[T1]{fontenc} \usepackage[utf8]{inputenc} \usepackage{relsize} \usepackage{times} \usepackage{url} \usepackage{latexsym} \usepackage{graphicx} \usepackage{colortbl} \usepackage{color} \usepackage{caption} \usepackage{pgfplots, pgfplotstable} \usetikzlibrary{arrows} \usepackage{amsmath} \usepackage{multirow} \usepackage{booktabs}  \usepackage{filecontents} \definecolor{OgAns}{rgb}{0, 0.8, 0.4}  \begin{filecontents}{testdata.dat} 1   30 2   44 4   26 3   39 \end{filecontents}  \begin{document} \begin{figure} \begin{tikzpicture}         \begin{axis}[                ybar stacked,             ymin=0,             ymax=100,             bar width=5pt,             legend style={at={(0.35, -0.4)},anchor=south west},             legend columns=-1                       ]           \addplot[ybar,fill=OgAns]  file {testdata.dat};          \end{axis}     \end{tikzpicture} \end{figure}   \end{document} Given a real positive sequence $\{a_n\}$ such that $\sum_{i=1}^n a_i$ converges.  Prove that there exists a real sequence $\{c_n\}$ monotonically increasing to $\infty$ such that $\sum_{i=1}^\infty a_nc_n$ converges.  What I have done:  I have written $$\sum_{i=1}^{N+1}a_ic_i=c_{N+1}\sum_{i=1}^{N+1}a_i-\sum_{i=1}^N(c_{n+1}-c_n)S_n$$  and then try to prove the convergence by the Cauchy criterion. However my evaluation is still not good enough. I'm really stuck. Hahn-Banach Theorem for separable spaces without Zorn's Lemma 16.04 Upgrade: Failed to fetch Empty files can't be valid archives Benefit of date in URL segments I was curious about the date in URL segments, is there any SEO point of benefit? For example the following URLs:  http://techcrunch.com/2013/06/18/netflix-will-launch-in-the-netherlands-later-this-year-as-its-international-expansion-slows/ http://tips-for-new-bloggers.blogspot.in/2007/02/adding-favicon-icon-to-blogger-url.html  https://community.jivesoftware.com/blogs/jivespace/2013/05/15/eol-of-the-html5-ipad-web-app   In all the above URLs there is date segment. The information of those blogs can be fetched via blog stub like "netflix-will-launch-in-the-netherlands-later-this-year-as-its-international-expansion-slows" so why to use that date part? Give each site a parallel site for polling, recommendations and subjective-ish stuff What is the best algorithm for overriding GetHashCode? In .NET, the  is used in a lot of places throughout the .NET base class libraries. Implementing it properly is especially important to find items quickly in a collection or when determining equality.  Is there a standard algorithm or best practice on how to implement GetHashCode for my custom classes so I don't degrade performance? Unable to resume after suspend I'm using Ubuntu 12.04 &amp; everything about it is great except that it just won't resume after suspend. I've tried the suggestions listed in this thread  but none worked. I'm using Lenovo Y510P. I'm new to Ubuntu, or unix for that matter, so please let me know if there's anymore  information that I should be posting. Thanks.  update  I tried the suggestion by Dima on this post . The machine wakes up from hibernation but immediately the screen gives a screen like this     Wondering if this is an issue with the graphics card or something. Let $\displaystyle f$ be differentiable, $\displaystyle f(x)=0$ for $|x| \geq 10 $ and $g(x)=\sum_{k \in \mathbb Z}f(x+k).$ Identify the plane defined by $|z-2i| = 2|z+3|$ I tried:  $$|z-2i| = 2|z+3| \Leftrightarrow \\ |x+yi-2i|=2|x+yi+2|\Leftrightarrow \\ \sqrt{x^2+(y-2)^2}=\sqrt{4((x+2)^2+y^2)} \Leftrightarrow \\ \sqrt{x^2+y^2-4y+4} = \sqrt{4x^2+24x+36+4y^2} \Leftrightarrow \\ x^2+y^2-4y+4 = 4x^2+24x+36+4y^2 \Leftrightarrow \\ y^2-4y-4y^2=4x^2+24x+36+x^2 \Leftrightarrow \\ -3y^2-4y=5x^2+24x+26 \Leftrightarrow \\ ???$$  What do I do next? Let $X$ be topological space and $A\subseteq X$.   If $E\subseteq X$ is connected, $E\cap A\neq\emptyset$ and $E\cap(X\setminus A)\neq\emptyset$ then $E\cap\partial A\neq\emptyset$.  Assume that $E\cap\partial A=\emptyset$.  If $E\cap\partial A=\varnothing$ then $E=(E\cap\text{int}(A))\cup(E\cap\text{ext}(A))$, i.e. a union of disjoint sets open in $E$.   Also the sets are nonempty because: $E\cap\text{int}(A)=E\cap A\neq\varnothing$ and $E\cap\text{ext}(A)=E\cap A^c\neq\varnothing$.  This contradicts that $E$ is connected.  Fixed. Can't save settings after copying site to new server After booting I ran systemd-analyze blame and here are the results:       21.596s systemd-journal-flush.service      18.658s dev-sda8.device      15.099s dev-loop33.device      15.034s dev-loop19.device      15.012s dev-loop34.device      14.989s dev-loop21.device      14.877s dev-loop15.device      14.866s dev-loop26.device      14.773s dev-loop27.device      14.684s dev-loop30.device      14.677s dev-loop32.device      14.649s dev-loop35.device      14.590s dev-loop25.device      14.267s dev-loop23.device      14.192s dev-loop24.device      14.156s dev-loop29.device      14.133s dev-loop16.device      14.065s dev-loop31.device      14.059s dev-loop28.device      13.821s dev-loop20.device      13.531s dev-loop22.device      13.495s dev-loop14.device      13.364s dev-loop18.device   What are these dev-loopxx.device (xx denotes numbers) services and why are they taking so much time? Are they related to the mounting of snaps? Can I reduce the boot time by disabling them? I'm running Ubuntu 18.04 alongside Windows 10. Format SQL in SQL Server Management Studio In Visual Studio &amp; other IDEs, you can easily auto format your code with a keyboard shortcut, through the menu, or automatically as you type.  I was wondering if there is yet a way to enable this standard feature in SQL Server Management Studio?  I'm working with some large-ish stored procs that are a mangled mess of poorly formatted SQL and it'd be nice if I could just go "Select All -> Format SQL" What are the drawbacks with using UUID or GUID as a primary key? I would like to build a distributed system. I need to store data in databases and it would be helpful to use an  or a  as a primary key on some tables. I assume it's a drawbacks with this design since the UUID/GUID is quite large and they are almost random. The alternative is to use an auto-incremented INT or LONG.  What are the drawbacks with using UUID or GUID as a primary key for my tables?  I will probably use Derby/JavaDB (on the clients) and PostgreSQL (on the server) as DBMS.