Isaacs 6.13: If $\textbf{Z}(A_n)&gt;1$, then $n=3$.  I am unsure where to begin. I know that $\textbf{Z}(A_n)$ is the center of $A_n$. If we assume that $\textbf{Z}(A_n)&gt;1$, then the center is not the trivial center (i.e. it doesn't only contain the identity element.  But how do I proceed from  there?
I need to prove that the center of $A_n$ is trivial for $n \geq 4$.  $Z(A_3) = A_3$, since $A_3 = \mathbb{Z}/3\mathbb{Z}$ is commutative.  One idea is two use "counting" technique. First of, all we count cojugacy classes of even permutations in $S_n$. It's the classes correspodning to the types $[\lambda_1, ..., \lambda_r]$ where $n \equiv r \mod 2$  Now, the conjugacy class in $S_n$ correspoding to the type $[\lambda_1, ..., \lambda_r]$ splits into two equal-sized classes in $A_n$ if $\lambda_1, ..., \lambda_r$ are distinct odd numbers. If they aren't then it is preserved in $A_n$.  Then we can use "the counting formula"(  ) of elements in each "even" class of $S_n$ and divide one by $2$ if needed.  Still, I'm not sure if it can be done. Seems to be a lot of work there. Maybe there are other, he easier ways? Or, maybe, it is the way, in this case, I wouild appreciate any advices.
How to clip an Openlayers OSM Background to a Vector Boundary? Setting DIV width and height in JavaScript Find $\sum_{n=1}^\infty \frac{n^3}{n!}$. So I've been experiencing this in the last few days: sometimes when someone replies to my comment, and I reply to the reply, then the "add/show 1 more comment" link appears immediately after I post my comment. However, when I click it, it disappears and nothing is shown.  I don't think it's that OP is posting two comments of which he deletes the second one, because this has happened to me several times, even when I was replying within 2-3 seconds.  I do, however, believe that this is more or less the same bug as , except that I don't have to reload the page in order the link to be gone, and I also seem to be getting the erroneous behavior only if I reply to a comment which is itself a reply to an earlier comment of mine.  Unfortunately, I wasn't careful enough to take a screenshot last time this happened, but here's what I am talking about for those who like visual demonstration better: Proving set of interiors is open What is the difference between using a colon and a semi-colon to join two sentences? I've seen a few questions on this site relating to semi-colons, which I believe I correctly understand, but what I'm not as clear about is colons.  For example:     The man ate the apple; it tasted good.   Is this not correctly also written as:     The man ate the apple: it tasted good.   In this case, what is the difference between the two sentences? Can't execute terminal commands: "the command 'sudo' is available in '/usr/bin/sudo'" If user input is inserted without modification into an SQL query, then the application becomes vulnerable to , like in the following example:  $unsafe_variable = $_POST['user_input'];   mysql_query("INSERT INTO `table` (`column`) VALUES ('$unsafe_variable')");   That's because the user can input something like value'); DROP TABLE table;--, and the query becomes:  INSERT INTO `table` (`column`) VALUES('value'); DROP TABLE table;--')   What can be done to prevent this from happening? Institute Name in @masterthesis Citations I need to cite master thesis and the lecture notes into a document. But the problem with @masterthesis and @unpublished citations is they don't print an institution name in the pdflatex output of bibliography.   `\documentclass{article}   \usepackage{filecontents}  \usepackage[ backend=biber, style=numeric, sortlocale=de_DE, natbib=true, url=true,  doi=true, eprint=false ]{biblatex} \addbibresource{bibliography.bib}  \begin{filecontents}{bibliography.bib} @masterthesis{bhgt, title={Bike Model}, author={Fris}, year={2016}, school={University A}, address={XXX} } @unpublished{Map, title = {Electrical Engineering Book}, author={Koch}, howpublished = {Lecture Notes}, Institution= {University B}, year = {2015} } } \end{filecontents}  \begin{document}  Citation Test:  \cite{bhgt} and \cite{Map} \printbibliography  \end{document}`   I get this bibliography output:         [1] Fris. Bike Model. XXX, 2016.   [2] Koch. “Electrical Engineering Book”. Lecture Notes. 2015.       Any suggestions, how we can also print an institute name? Proving that $\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots + \frac{1}{n(n+1)} = \frac{n}{n+1}$ How would we go about proving that $$\frac{1}{1\cdot 2} + \frac{1}{2\cdot 3} + \frac{1}{3 \cdot 4} +\ldots +\frac{1}{n(n+1)} = \frac{n}{n+1}$$ Big Greeks and commutation About the localization of a UFD I was wondering, is the localization of a UFD also a UFD?    How would one go about proving this? It seems like it would be kind of messy to prove if it is true.  If it is not true, what about localizing at a prime? Or what if the UFD is Noetherian? Suppose we have two short exact sequences in an abelian category $$0 \to A \mathrel{\overset{f}{\to}} B \mathrel{\overset{g}{\to}} C \to 0 $$ $$0 \to A&#39; \mathrel{\overset{f&#39;}{\to}} B&#39; \mathrel{\overset{g&#39;}{\to}} C&#39; \to 0 $$ and morphisms $a : A \to A&#39;, b : B \to B&#39;, c : C \to C&#39;$ making the obvious diagram commute. The snake lemma states that there is then an exact sequence $$0 \to \ker a \to \ker b \to \ker c \to \operatorname{coker} a \to \operatorname{coker} b \to \operatorname{coker} c \to 0$$ where the morphisms between the kernels are induced by $f$ and $g$ while the maps between the cokernels are induced by $f&#39;$ and $g&#39;$.  It is not hard to show that the morphisms induced by $f, g, f&#39;, g&#39;$ exist, are unique, and that the sequence is exact at $\ker a, \ker b, \operatorname{coker} b, \operatorname{coker} c$. With the use of a somewhat large diagram shown , we can even construct the connecting morphism $d : \ker c \to \operatorname{coker} a$. However, I'm stuck showing exactness at $\ker c$ and $\operatorname{coker} a$. I thought Freyd might have had an element-free proof in his book, but it turns out he proves it by diagram chasing and invoking the Mitchell embedding theorem [pp. 98–99]. Is there a direct proof? How to modify itemize vertical spacing within blocks in beamer? I have noticed that when I have "itemize" environments defined within blocks in beamer, that they introduce more vertical spacing between the top and the bottom of the blocks, than vertical text. Since I have a slide with some text filled blocks and directly after, a slide with only "itemize" holding blocks, I would like the spacing between the itemized text and the block to be the same as for regular text.   On  I have read about "squeeze" option, but it doesn't seem to work...   Additional info:  The problem is in the additional vertical space that is included by the "\itemize" environment. I have found  on how to remove vertical spacing from the itemize list, and I am using this approach. However, it is a brute force approach. What I would like to know if it is possible to have the same vertical space between the end of the itemize list and the bottom of a block, as the one set between the normal text and the bottom of the block.   MWE:   \documentclass[10pt]{beamer} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{exscale} \usepackage{epsfig} \usepackage{fancybox} \usepackage{lmodern} \usepackage[T1]{fontenc} \usepackage{overpic}  \usepackage{multimedia}  \usetheme{default} \usecolortheme{crane}  % -- Modifications that enable the footline and set black color for the fonts  \setbeamertemplate{footline}{%     \begin{beamercolorbox}[wd=\paperwidth,ht=2.25ex,dp=1ex,right, rightskip=1mm, leftskip=1mm]{titlelike}         \textcolor{black}{             %\inserttitle\hfill\insertauthor\hfill\insertframenumber%             Something\hfill Something Else \hfill\insertframenumber%         }     \end{beamercolorbox} }  \setbeamertemplate{itemize items}[square]  \setbeamercolor{item}{fg=black} \setbeamercolor{frametitle}{fg=black} \setbeamercolor{block title}{fg=black} \setbeamercolor{title}{fg=black} \setbeamercolor{block body}{bg=white}  % END Modifications  \begin{document}   \begin{frame}   \frametitle{text blocks}     \begin{block}{block}         block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   blockk     \end{block}{block}      \begin{block}         block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block     \end{block}     \begin{block}{block}         block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block   block     \end{block} \end{frame}  \begin{frame}   \frametitle{itemize blocks}     \begin{block}{item}         \begin{itemize}             \item item              \item item              \item item         \end{itemize}     \end{block}     \begin{block}{item}         \begin{itemize}             \item item              \item item              \item item         \end{itemize}     \end{block}     \begin{block}{item}         \begin{itemize}             \item item              \item item              \item item         \end{itemize}     \end{block}   \end{frame}  \end{document} Most efficient way to randomly "sort" (Shuffle) a list of integers in C# How is Clan War Map position determined? What happens to the second Marty at the end of Back to the Future? In  (the original movie), Marty comes back from 1955 10 mins early to prevent Doc from being shot (which he fails at). He then rolls up to the Lone Pine Mall and sees himself hit 88 mph and shoot back to 1955 (again). Marty then carries on with this life as usual. What happens to the Marty that goes back to 1955 at the end of the film?  This obviously is a recursive problem where it would create another timeline and Marty would keep coming back 10 mins early to view another Marty zip back to 1955? Difference between standard error and standard deviation I'm struggling to understand the difference between the standard error and the standard deviation. How are they different and why do you need to measure the standard error? How to send a header using a HTTP request through a cURL call? I wish to send a header to my Apache server on a Linux box.  How can I achieve this via a cURL call? What color can lightsabers be? I just saw the fan film  and the Force-sensitive bounty-hunter, Korbain Thor, wielded an ochre lightsaber (or at least that's what  described it as).   I'm just wondering if ochre is a canon color for a lightsaber?