By definition, if $H$ is characteristic subgroup of $G$, then every automorphism of $G$ restricted on $H$ is an automorphism of $H$.  From these two questions   we know it's generally false that if $H$ is characteristic subgroup of $G$, given $\varphi \in \text{Aut}(H)$, then there exists $\psi\in\text{Aut}(G)$ s.t. $\psi|_H=\varphi$. Example from $A_3$ and $S_3$ shows even if such a $\psi$ exists, it may not be unique.  My question:  Suppose $H$ is subgroup of $G$, what's the sufficient condition s.t. $\forall \varphi \in \text{Aut}(H)$, $\exists\ \psi\in\text{Aut}(G)$ s.t. $\psi|_H=\varphi$?  Background:  This question comes from studying $\text{Aut}(S_n)$ and $\text{Aut}(A_n)$.  If $n\geqslant 4$ and $n \neq 6$, then $\operatorname{Aut}(A_n)\cong\operatorname{Inn }(S_n)\cong\operatorname{Aut}(S_n)\cong S_n$.  The existence and uniqueness is true for $A_n$ and $S_n$($n\geqslant 4$ and $n \neq 6$), while uniqueness is not true for $A_3$ and $S_3$.  To move this question out of the unanswered list, I put related materials in answer.
$\DeclareMathOperator{\Char}{char}$$\DeclareMathOperator{\Aut}{Aut}$ Let $G$ be a finite group, $H$ a subgroup, such that $H \Char G$. If $\phi \in \Aut(H)$ is there an automorphism $\widehat{\phi} \in \Aut(G)$ such that $\widehat{\phi}|_H$ = $\phi$?  In words, if $\phi$ is an automorphism of $H$, is there necessarily an automorphism $\widehat{\phi}$ of $G$ where $\widehat{\phi}$ maps the elements in $H$ to $H$ in exactly the same way?  I am thinking this is false (consider the non-finite example of $\bar{Q}$ and $\mathbb{C}$), but would like a counter example (for the finite case).
How can I efficiently drive an LED? I understand that I can not connect an LED directly to a battery because it will draw too much current. Thus, there must be something else in the circuit to limit the current. What options are there? Are some methods more efficient than others? quotation mark (quotation sign) xelatex + polyglossia + csquotes the problem I have is the "?" that shows instead of appropriate quotation marks (signs) when citing/quoting text in foregin language. I have searched manuals for the packages such as polyglossia or csquotes.   Unfortunately I haven't found a command that would allow me to declare quotation marks (signs) for the languages I am intending to use in my doc.   In the plain text I could of course use the ,, or '' to mimic the behavior of enquote but the bibliographic entries ale also affected by this unpleasant phenomena :(  % !TEX TS-program = arara % !TEX encoding = UTF-8 Unicode  % arara: xelatex: { shell: true } % arara: biber % arara: xelatex: { shell: true }  \documentclass[12pt]{article}  \usepackage[no-math]{fontspec} \defaultfontfeatures{Ligatures=TeX} \newfontfamily\greekfont[Script=Greek,Scale=MatchUppercase]{Linux Libertine O} \newfontfamily\cyrillicfont[Script=Cyrillic,Scale=MatchUppercase]{Linux Libertine O}  \usepackage{polyglossia} \setdefaultlanguage{polish} \setotherlanguages{english,latin,greek,russian,german}  \usepackage[strict=false,autostyle=true,english=american,german=guillemets]{csquotes}  \PassOptionsToPackage{%         natbib=true,         style=authoryear-comp,         hyperref=true,         backend=biber,         maxbibnames=99,         firstinits=true,         uniquename=init,         maxcitenames=1,         citetracker=true,         parentracker=true,         backref=true,         backrefstyle=two,             }   {biblatex} \usepackage{biblatex}  \usepackage{filecontents} \begin{filecontents}{bibliography.bib} @article{Author2014,     author      = {Joe, Smith},     title       = {Publication title in polish language},     journal     = {Journal of Journals},     pages       = {1},     year        = {2014},     month       = {March}     } \end{filecontents}  \addbibresource{bibliography.bib}  \begin{document}      \noindent{}{\enquote{\languagename{} \today}}\\     \textenglish{\enquote{\languagename{} \today}}\\     \textgerman{\enquote{\languagename{} \today}}\\     \textlatin{\enquote{\languagename{} \today}}\\     \textgreek[variant=ancient]{\enquote{\languagename{} \today}}\\     \textrussian{\enquote{\languagename{} \today}}\\     \textpolish{\enquote{\languagename{} \today}}\\     \cite{Author2014}      \printbibliography  \end{document} Limit of the sequence $a_n=\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}$ ffmpeg synchonization of libfdk_aac (or libfacc) and libx264 How to synchronize libfdk_aac and libx264 streams?  ffmpeg -i pipe_libfdk_aac.fifo -i pipe_libx264.fifo -c copy -f flv flv_stream_pipe_or_rtmp.output   I'm always receiving the same asych in all cases: saving file on file system, rtmp-using or raw pipe out. Is it possible to syncronize them?   ffmpeg version N-66924-gaba61b2 Copyright (c) 2000-2014 the FFmpeg developers built on Jan 15 2015 12:50:29 with gcc 4.6 (GCC) 20120106 (prerelease) configuration: --arch=arm --cpu=cortex-a8 --target-os=linux --enable-runtime-cpudetect --prefix=/data/data/info.guardianproject.ffmpeg/app_opt --enable-pic --disable-shared --enable-static --cross-prefix=/home/qwerty/work/android-ndk-r10b86/android-ndk-r10b/toolchains/arm-linux-androideabi-4.6/prebuilt/linux-x86/bin/arm-linux-androideabi- --sysroot=/home/qwerty/work/android-ndk-r10b86/android-ndk-r10b/platforms/android-3/arch-arm --extra-cflags='-I/home/qwerty/x264 -I/home/qwerty/ffmpeg_build/mp3/build/include -I/home/qwerty/faac/output/include -I/home/qwerty/ffmpeg_sources/fdk-aac/build/include -I/usr/local/include/ -mfloat-abi=softfp -mfpu=neon' --extra-ldflags='-L/home/qwerty/x264 -L/home/qwerty/ffmpeg_build/mp3/build/lib -L/home/qwerty/ffmpeg_sources/fdk-aac/build/lib -L/home/qwerty/faac/output/lib -L/usr/local/lib/ -lrtmp -lpolarssl' --enable-version3 --enable-gpl --enable-nonfree --disable-doc --enable-yasm --enable-decoders --enable-encoders --enable-muxers --enable-demuxers --enable-parsers --enable-protocols --enable-filters --enable-avresample --enable-librtmp --enable-libfaac --enable-libfdk_aac --enable-libmp3lame --disable-indevs --enable-indev=lavfi --disable-outdevs --enable-hwaccels --enable-ffmpeg --disable-ffplay --disable-ffprobe --disable-ffserver --disable-network --enable-libx264 --enable-zlib --enable-muxer=md5 libavutil      54. 10.100 / 54. 10.100 libavcodec     56.  8.100 / 56.  8.100 libavformat    56.  9.100 / 56.  9.100 libavdevice    56.  1.100 / 56.  1.100 libavfilter     5.  1.106 /  5.  1.106 libavresample   2.  1.  0 /  2.  1.  0 libswscale      3.  1.101 /  3.  1.101 libswresample   1.  1.100 /  1.  1.100 libpostproc    53.  3.100 / 53.  3.100 How do I create an 8 sided die from one object? Simply put, is there a way to make a cube balance solely on one point/vertex?  I'm struggling. I do not want to have to join two objects, nor have to extrude an object. I just want to have a simple eight-sided die created with one mesh. Explanation of unpredictable behaviour of tee I encountered a behaviour that I don't understand while testing a script that sums the outputs from repeated executions of a program. To reproduce it create the text files out, which represents the output of my program, and sum, the file that holds the sum of the values returned on previous executions and which starts out as a copy of out,  cat &gt; out &lt;&lt; EOF 2 20 5 50 EOF cp out sum   The strange thing happens on running  paste out sum | awk '{$1 += $3; $2 += $4; NF = 2; print}' | tee sum   several times (15-20 times might be needed). Each time it runs, this command should add to the values in sum the corresponding values in out and write the results back to sum. What I get is that it works an unpredictable number of times, then sum reverts back to  2 20 5 50   I have later learned that  and solved the issue using a temporary file, still, this behaviour baffles me:   why does … | tee sum work at all (even if only for a limited number of iterations), while … &gt; sum never overwrites sum? why doesn't it work a predictable number of times? What are Alternative's "some" and "many" useful for? returning a local variable from function in C Idiom for explaining something to someone who clearly knows the topic (probably better than you) Do resistances stack? If a creature had resistance to spells, and then cast fire resistance on it, would it take only 1/4 damage from all fire spells? Why are conference papers so important in computer science (CS)? From my understanding many CS conferences are highly selective and the papers are peer-reviewed. The CS field also seems to publish traditional journal articles. I do not understand the difference between the two and in particular I am curious about the advantages of having two different publication streams. The type of things I am curious about are:   What is the difference between a paper published in a top CS conference and a top specialist journal? Is one more prestigious than another and if so how do they line up? Why not just publish conference papers as a special issues of a journal? Creating a DateTime in a specific Time Zone in c# I'm trying to create a unit test to test the case for when the timezone changes on a machine because it has been incorrectly set and then corrected.  In the test I need to be able to create DateTime objects in a none local time zone to ensure that people running the test can do so successfully irrespective of where they are located.  From what I can see from the DateTime constructor I can set the TimeZone to be either the local timezone, the UTC timezone or not specified.  How do I create a DateTime with a specific timezone like PST? Node for setting render colour, regardless of lighting I'm trying to make a material node setup to set final render colors, regardless of lighting conditions. Is such a setup possible? What's the 'Frequent' sort function? What is the function of the  sorting option on the Questions tab? Confidence interval around binomial estimate of 0 or 1 What is the best technique to calculate a confidence interval of a binomial experiment, if your estimate is that $p=0$ (or similarly $p=1$) and sample size is relatively small, for example $n=25$? Commenting out large sections To &quot;comment out&quot; a line, I need to insert a % at the beginning of the line (so that the line will not be compiled). Is there a way to comment out a large section without having to manually place % in front of each line? Let us view our deleted comments how i evaluate this integral with integration by substitution $x=\frac{\pi}{2}-t$  $$\int _0^{\frac{\pi }{2}}\:\frac{\sqrt[3]{\sin^2\left(x\right)}}{\sqrt[3]{\sin^2\left(x\right)}+\sqrt[3]{\cos^2\left(x\right)}}dx$$ $|p(z)| \leq M$ for $|z| \leq 1$ Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$ Let $p(z)= \sum_{k=0}^n a_k z^k$ ,  $a_n \neq 0$ , be a polynomial of degree $n$ such that $|p(z)| \leq M$ for $|z| \leq 1$.  Show that $|p(z)| \leq M|z|^n$ for $|z| \geq 1$  This was an exam question. Part (a) of this question asked the converse: if $|p(z)| \leq M|z|^n$ and $|z|$ is sufficiently large, then $p(z)$ is a polynomial. This question came straight out of Ahlfors' Complex Analysis, which I solved by showing, using the Cauchy Estimate,  $|p^n (z)|  \leq \frac{n!M|z|^k}{R^n} = 0$ as $R \to \infty$.   I would like to work backwards with this.   May I do this, or similar:   $p(z) = \frac{1}{2\pi i} \int_{c} \frac{f(\zeta)d\zeta}{\zeta-z}$  $\Rightarrow  |p(z)| \leq \frac{2\pi M|z|^n}{1(2\pi)}= M|z|^n$ Manually Installed new Python to /usr/bin, how to get Repo version back? Yes, I'm a moron, and I tried to manually install a higher point release of python 2.7.16 to /usr/bin/  (i did ./configure --prefix=/usr) on my Ubuntu 16.04 machine. It seems to work fine EXCEPT for packages that depended on python, like apt-listchanges, which fails with messages like ImportError: No module named apt_pkg  there's no make uninstall available, I've tried manually removing files, and still when I try apt --reinstall install python, Nothing gets overwritten and the python executable still gives the new 2.7.16 version...   Any idea how to get the official repository version back??