If |G/H|=4 and G/H  is not cyclic then G is union of three proper subgroups. I understand that there are three subgroups of order two in the G/H group. Consequently, in the group G there are three subgroups A, B, C of index two which in them pass under the homomorphism. But how to prove that G is a union of the subgroups A, B, and C
Let $G$ be a finite group and $N$ a normal subgroup of $G$ of index $4$. Prove that $G$ has a subgroup of index $2$ (hence normal) and if $G/H$ is not cyclic then $G$ is equal to the union of three proper subgroups.   I understand that $G/H$ is isomorphic to $\mathbb{Z}_4$ or $\mathbb{Z}_2\times \mathbb{Z}_2$. Let $K^*$ be a subgroup of $G/H$ of order $2$. Then $K^*\cong K/H$ for some $K$ such that $H$ is normal in $K$ and hence $[G:K]=[G/H:K/H]=4/2=2$.  If $G/H$ is not cyclic then $G/H\cong \mathbb{Z}_2\times \mathbb{Z}_2$. I don't know how to prove $G$ is the union of three subgroups. Could it be true that as $G/H$ is the union of three proper subgroups $K_1^*,K_2^*,K_3^*$ then the corresponding groups $K_1,K_2,K_3$ have union equal to $G$? Would anyone give me a hint to prove this?  Thank you.
Best font for programming? Problem with literate and breaklines=true in listings package As the title says, the option breaklines=true seems to have an undesired interaction when using literate; in the following example the closing parenthesis doesn't get colorized   \documentclass{article} \usepackage{xcolor} \usepackage{listings}  \lstset{   literate=     {)}{{\textcolor{red}{)}}}{1}     {(}{{\textcolor{red}{(}}}{1},   breaklines=true, }  \begin{document}  \begin{lstlisting} () \end{lstlisting}  \end{document}     Commenting out the breaklines=true option produces the desired result. What is producing this odd behaviour and how can it be prevented? Spacing changes when using unicode-math range feature. Why? You can use the range feature of unicode-math to set the math font for a specific unicode slots or the macro accessing the glyph. I tried this with the sum sign &amp; Latin Modern, see my example:  \documentclass{scrartcl}  \usepackage{amsmath} \usepackage{unicode-math} % use unicode in math &amp; setup math fonts \setmathfont{Asana Math} %\setmathfont[range=\sum]{Latin Modern Math} % substitute Sum sign  \begin{document}  Notice the different sum sign $\sum$, as intended.   Notice also, the difference in spacing, not intended: \begin{equation}     \begin{bmatrix}         0           &amp; \frac{1}{2}   &amp; \frac{2}{3} \\         \frac{3}{4} &amp; 0             &amp; \frac{4}{5} \\         \frac{5}{6} &amp; \frac{6}{7}   &amp; 0     \end{bmatrix}     \begin{pmatrix}         A \\         B \\         C     \end{pmatrix} \end{equation} \end{document}   However, not only the sum sign, but also the spacing in the matrices changes. With Latin Modern commented out (as in the example):  With Latin Modern active (remove % before relevant line):  Why is that? BTW, I'm using Lualatex from TeXLive 2012. Is list::size() really O(n)? Recently, I noticed some people mentioning that std::list::size() has a linear complexity. According to  , this is in fact implementation dependent as the standard doesn't say what the complexity has to be. The comment  says:     Actually, it depends on which STL you   are using. Microsoft Visual Studio V6   implements size() as {return (_Size);   } whereas gcc (at least in versions   3.3.2 and 4.1.0) do it as { return std::distance(begin(), end()); } The   first has constant speed, the second   has o(N) speed    So my guess is that for the VC++ crowd size() has constant complexity as Dinkumware probably won't have changed that fact since VC6. Am I right there? What does it look like currently in gcc? If it is really O(n), why did the developers choose to do so? "Geometric" or "Geometrical"? I have read the excellent answers to  - my question is specifically about usage.   Is there a best practice for deciding between the variants geometric and geometrical in writing? A  shows that geometrical was more popular until about 1965, when they switched. What grammatical rules, if any, would be relevant here? How to solve and see resolution of $13^{53} \pmod 7$ using Fermat little Theorem? Prove every odd integer is the difference of two squares I know that I should use the definition of an odd integer ($2k+1$), but that's about it.  Thanks in advance! What is this story about 20 y.o. inexperienced "kids" fighting against a stronger foe in space? Submitting a form by pressing enter without a submit button Well I am trying to submit a form by pressing enter but not displaying a submit button. I don't want to get into JavaScript if possible since I want everything to work on all browsers (the only JS way I know is with events).  Right now the form looks like this:  &lt;form name="loginBox" target="#here" method="post"&gt;     &lt;input name="username" type="text" /&gt;&lt;br /&gt;     &lt;input name="password" type="password" /&gt;     &lt;input type="submit" style="height: 0px; width: 0px; border: none; padding: 0px;" hidefocus="true" /&gt; &lt;/form&gt;   Which works pretty well. The submit button works when the user presses enter, and the button doesn't show in Firefox, IE, Safari, Opera and Chrome. However, I still don't like the solution since it is hard to know whether it will work on all platforms with all browsers.  Can anyone suggest a better method? Or is this about as good as it gets? What is the technical term for when people thinks the actions of the few represent the many.  Like when people thinks everyone in a specific group behaves a certain way cause of how a few people in the group behave, including non stereotypical behaviours. I thought it was "generalisation" but I couldn't find any articles using that term. Few days ago I got sample of toroidal core from my friend. I don't know any technical details of core just know OD-10cm, ID-6cm, height- 5cm and wet about 1.845 kg(M4 grade CRGO). I want to make toroidal transformer using this core but how to calculate core area and maximum VA capacity of core.I take some photos and attach. Incompatibility between TikZ and babel In my preamble, you'll find:  \renewcommand{\baselinestretch}{1} \renewcommand{\familydefault}{\sfdefault} \usepackage[sfmath]{kpfonts} \usepackage[mathrmOrig]{sfmath} \usepackage{amsmath} \usepackage{amssymb} \usepackage{amsfonts} \usepackage{animate} \usepackage{anysize} \usepackage{arcs} \usepackage{array} \usepackage[scale=0]{background} \usepackage{blindtext} \usepackage{calc} \usepackage{cancel} \usepackage{chngcntr} \usepackage{CJK} \usepackage{CJKspace} \usepackage{CJKpunct} \usepackage{colortbl} \usepackage{enumitem} \usepackage{epstopdf} \usepackage{etoolbox} \usepackage{fancybox} \usepackage{fancyhdr} \usepackage{float} \usepackage{framed} \usepackage{graphicx} \usepackage{hyphenat} \usepackage{ifthen} \usepackage{latexsym} \usepackage{lmodern} \usepackage{multicol} \usepackage{multirow} \usepackage{mwe} \usepackage{pgffor} \usepackage{rotating} \usepackage{tabularx} \usepackage{textcomp} \usepackage{tgheros,tgtermes,tgcursor} \usepackage{tikz} \usepackage{times} \usepackage{tipa} \usepackage{varwidth} \usepackage{wasysym} \usepackage{wboiboites} \usepackage[abs]{overpic} \usepackage[breaklinks,colorlinks=true,linkcolor=red,citecolor=red, urlcolor=blue]{hyperref} \usepackage[framemethod=TikZ]{mdframed} \usepackage[breakable, theorems, skins]{tcolorbox,empheq} \usepackage[default]{gillius} \usepackage[document]{ragged2e} \usepackage[latin1]{inputenc} \usepackage[spanish]{babel} \usepackage[T1]{fontenc} \usepackage[total={18.59cm,26.5cm},left=1.5cm,top=2.5cm]{geometry} \usetikzlibrary{matrix,arrows, positioning,shadows,shadings,backgrounds,calc, shapes, tikzmark} \usetikzlibrary{shadows.blur} \usetikzlibrary{shapes.symbols} \usetikzlibrary{patterns,fadings} \tcbuselibrary{skins,breakable,listings,theorems}    I'm trying to draw an arrow for X axis and Y axis.   \begin{tikzpicture} \draw[yellow!10, fill=blue!5](-1, -1) rectangle (7, 5); \draw[dotted] (-0.75,-0.75) grid (6.75,4.75); \node[below] at (-0.25,0){$O$}; \draw[-&gt;,black, thick] (-0.75,0)--(6.75,0); \end{tikzpicture}   But I get this error:    Additionally, if I remove -> from the options of \draw the tikz picture compiles normally. Placement of question mark when terminal quote has no question mark but sentence does Possible Duplicate:          If I'm quoting something someone said and the quote does not have a question mark, but I want to end the sentence with a question mark, would it be     Are you sure she said "Spicy?"   or       Are you sure she said "Spicy"? Consider the following example:  \documentclass{article} \newtheorem{Theorem}{Theorem} \begin{document} \begin{Theorem} On the other hand, this one has only one claim. \end{Theorem} \begin{Theorem} \begin{enumerate} \item First claim \item Second claim \end{enumerate} \end{Theorem} \end{document}   Typesetting it results in:    Notice there is extra horizontal space before the first item in the list of the second theorem.     Is there a sensible way to get rid of that?   Ideally, the solution would work with amsthm, thmtools, enumitem.  N.B.: this is related to  but not quite the same. Proving $\sup B=\inf A$ Let $A$ be a nonempty and bounded below, and define $B= \{b\in \mathbb{R}: b$ is a lower bound for $A\}$. Show sup$B$ = inf$A$.   So far I have: Let $A$ be nonempty and bounded below. This implies $\exists l \in A, \forall a \in A, l\leq a$ and $l =$ inf$A$. Let $B = \{b\in \mathbb{R}: b$ is a lower bound for $A\}$ and let $M=$ sup$B$. This implies $\exists b\in B, \forall b \in B, b\leq M$. By definition of $B$, we know inf$A \in B$. Since $b\leq$ sup$B$, with sup$B$ the largest element in $B$, we have inf$A \leq M =$ sup$B$, so inf$A \leq$ sup$B$.  I realize I have to prove that sup$B \leq$ inf$A$ to get that sup$B$=inf$A$. I am having trouble doing so.   Part B: Use (a) to explain why there is no need to assert that greatest lower bounds exist as part of the Axiom of Completeness. optexp environment error when compiling with XeLaTeX I have troubles with optexp environment. There is a code  \documentclass[pstricks,margin=12pt]{standalone}  \usepackage{pst-optexp}  \begin{document} \begin{pspicture}(3.2,5) \pnodes(0,3){A}(1,3){BS}(3,3){M1}(1,5){M2}(1,1){PD} \psset{mirrortype=extended, mirrordepth=0.2}  \begin{optexp} \beamsplitter[bsstyle=plate, compname=BS](A)(BS)(PD) \mirror[compname=M1](BS)(M1)(BS) \mirror[compname=M2, variable](BS)(M2)(BS) \lens[compname=L](BS)(PD) \optdetector[compname=Det, dettype=diode](BS)(PD) \addtopsstyle{Beam}{beamwidth=0.2, fillstyle=solid,fillcolor=green!20!white} \drawwidebeam(A){BS}{M1}{BS}{M2}{BS}{L}{Det} \end{optexp} \end{pspicture}  \end{document}   and the result is this    and I get "MiKTeX GPL Ghostscript 9.19: Unrecoverable error, exit code 1" error. When I remove optexp environment, so  \documentclass[pstricks,margin=12pt]{standalone}  \usepackage{pst-optexp}  \begin{document} \begin{pspicture}(3.2,5) \pnodes(0,3){A}(1,3){BS}(3,3){M1}(1,5){M2}(1,1){PD} \psset{mirrortype=extended, mirrordepth=0.2}   \beamsplitter[bsstyle=plate, compname=BS](A)(BS)(PD) \mirror[compname=M1](BS)(M1)(BS) \mirror[compname=M2, variable](BS)(M2)(BS) \lens[compname=L](BS)(PD) \optdetector[compname=Det, dettype=diode](BS)(PD) \addtopsstyle{Beam}{beamwidth=0.2, fillstyle=solid,fillcolor=green!20!white} \drawwidebeam(A){BS}{M1}{BS}{M2}{BS}{L}{Det}  \end{pspicture}  \end{document}   the result is     which is correct but optical components are behind beam. I use XeLatex. What am I doing wrong? Which packages are missing? Thank you (example is from manual "Drawing optical experimental setups" where the code works correctly). How can I implement a comprehensive apa-style citation and bibliography I am really new to LaTeX and I need a comprehensive solution for my Thesis  I have this problem since two days and I couldn't find any solution in any other post.  I work with BibDesk on Mac  And I used Packages apacite and natbib  @article{Ha89, Author = {Peter M. Haas}, Date-Added = {2017-02-06 21:15:57 +0000}, Date-Modified = {2017-02-06 21:45:06 +0000}, Journal = {International Organization}, Lastchecked = {03/02/2010 12:54}, Number = {3}, Pages = {377-403}, Publisher = {Peter Haas}, Read = {0}, Title = {Do regimes matter? Epistemic communities and Mediterranean pollution control}, Urldate = { http://www.jstor.org/stable/2706652}, Volume = {43}, Year = {1989}}   this is a stripped version of my document:  \documentclass[11pt, oneside]{article}  \usepackage{lipsum} \usepackage[margin=1in, left=1.5in, includefoot]{geometry}   \usepackage{ragged2e} % REFERENCE PREAMBLE %\usepackage{url} \usepackage{apacite} \usepackage{natbib} %\usepackage{biblatex-apa}                                                   % Bullet preamble \renewcommand{\labelitemi}{$\bullet$} \renewcommand{\labelitemii}{$\diamond$} \renewcommand{\labelitemiii}{$\circ$} % graphics preamble \usepackage{graphicx}   %Allows you to import images \usepackage{float}      %Allows for control of float positions %HEADER AND FOOTER STUFF \usepackage{fancyhdr}  \pagestyle{fancy} \usepackage{graphicx}        \usepackage{amssymb} %Zeilenabstand 1,5 wenn aktiviert  \usepackage{setspace}\makeatletter\newcommand{\MSonehalfspacing}{\setstretch{1.44}\ifcase \@ptsize \relax\setstretch {1.448}\or\setstretch {1.399}\or\setstretch {1.433}\fi}\newcommand{\MSdoublespacing}{\setstretch {1.92}\ifcase \@ptsize \relax\setstretch {1.936}\or\setstretch {1.866}\or\setstretch{1.902}\fi}\makeatother\MSonehalfspacing %Zeilenabstand Code fertig \usepackage[hidelinks]{hyperref}   %allows for clickable reference %                       BEGIN OF DOCUMENT \citeA{source} %                       REFERENCES  \renewcommand{\sectionmark}[1]{\markright{\thesection\ #1}} \fancyhf{} \rhead{\fancyplain{}{Identifying the Regime}} \lhead{\fancyplain{}{List of References}} \cfoot{\fancyplain{}{\thepage}}  \setlength\bibitemsep{1.9\itemsep} \bibliography{/User/Desktop/Thesis/references/mybib.bib}                   \bibliographystyle{apa} \addcontentsline{toc}{section {\numberline{}References}                                                       \end{document}   I would like to have the bibliography entry include a retrieved from    I tried to deactivate hyperref – as suggested in another post – but it didn't work. I also tried to activate and deactivate natbib and apacite as well.   I would really appreciate any help... I'm building a child theme and it currently has a very simple &lt;head&gt; section in header.php:  &lt;head&gt;   &lt;meta charset="&lt;?php bloginfo( 'charset' ); ?&gt;"&gt;   &lt;meta name="viewport" content="width=device-width, initial-scale=1, maximum-scale=1"&gt;   &lt;title&gt;&lt;?php wp_title( '|', true, 'right' ); ?&gt;&lt;/title&gt;   &lt;link rel="profile" href="http://gmpg.org/xfn/11"&gt;   &lt;link rel="pingback" href="&lt;?php bloginfo( 'pingback_url' ); ?&gt;"&gt;   &lt;!--[if lt IE 9]&gt;   &lt;script type="text/javascript" src="&lt;?php echo get_template_directory_uri(); ?&gt;/js/html5shiv.min.js"&gt;&lt;/script&gt;   &lt;![endif]--&gt;    &lt;?php wp_head(); ?&gt; &lt;/head&gt;   I'm guessing wp_head() is responsible for the second &lt;title&gt; element (it appears just there in the final HTML), but other things I've read say this is impossible.  Should I be removing the &lt;title&gt; from my header.php, or should I be adding something to my functions to remove the title from wp_head() (eg. remove_action('wp_head', 'title') ?  Or should I be doing something else altogether? How to show equinumerosity of  the powerset of $A$ and the set of functions from $A$ to $\{0,1\}$ without cardinal arithmetic? When you edit a message and repost the contents of  its first revision, the repost is eaten up by chat.  Steps to reproduce:   Join any chat room Post something. Let's say you post abcd Edit that to something else. For example, --- Try to post abcd again. The green (pending) message will show up, but when the response is received, it disappears (the server spits out {"id":null,"time":null} as the response)   This happens more often than you think. Many a time I post something, only to realize that I want to use the magic separator (---) to separate it from a previous block. So I up-arrow, Ctrl-X, and type ---, and then I repost the cut text. Which gets eaten up by the server -- making a small change to the repost fixes things.