Let $G = \{e, a_1, a_2, \dots, a_n\}$ be a finite abelian group and let $S$ be the set of all the elements of $G$ which are not equal to their own inverse. The set $S$ can be divided up into pairs so that each element is paired off with its own inverse. Prove that $(a_1 a_2 \dots a_n)^2 = e$.   Every $x$ in $G$, outside $S$ abides by $x^2 = e.$   Let $a_i$ be in $S$. Since $G$ is abelian, we have  $a_ia_i^{-1} = a_i^{-1}a_i$  $e = a_i^{-1}a_i$  Since $a_i^{-1} = a_i$, $a_i^2 = e.$  Thus, $\dots a_i^2a_j^2\ldots a_n^2 = e.$  Does that make sense?  edit:  We need to prove $a_1a_2...a_na_1a_2...a_n = e$, right? Let $a_i$ be in $G \setminus S$. Then $(a_i)^2$ is $e$, is it? If we rearrange the elements on the left side of the equation above according to what belongs to $S$ and what belongs to $G \setminus S$ and take the product of all the $(a_i)^2$, then every element belonging to $G \setminus S$ will  be cancelled in the right side of $a_1a_2...a_na_1a_2...a_n = e$. So, we only need to consider the elements in $S$. Suppose $a_j \in S$. Let $a_k$ be the inverse of $a_j$. We need to prove $(a_j)^2 \cdot (a_k)^2 = a_j \cdot a_j \cdot a_k \cdot a_k = e$. Since $S$ is abelian , $a_j \cdot a_j \cdot a_k \cdot a_k$ can be rearranged as $a_j \cdot a_k \cdot a_j \cdot a_k$ which is $e$.
Suppose $G=\{a_1,...,a_n\}$ is a finite abelian group, and let $x=a_1a_2\dotsm a_n$.   Prove that if there is more than one element of order $2$ then $x=e$.   What I've done so far: (#1 is just for illustration, There should be at least $3$ such elements)   $G$ has an even number of elements with order $&gt;2$ (They are paired so: $(a,a^{-1})$ ). And then there are $e$ and $\{b|o(b)=2\}$. Also, if $G$ has at least one element of order 2 then $|G|$ is even (Lagrange). Thus $|\{b|o(b)=2\}|$ must be odd. If there is exactly one element $b$ of order $2$ then $x=b$. That's because $G$ is abelian, and we can write: $x=e\cdot b\cdot(a_1 a_{1}^{-1}\dotsm a_n a_{n}^{-1})=b\cdot(e\dotsm e)=b$. If there are exactly $3$ such elements $a,b,c\in G$, then, as shown above, $x=abc$ and thus $x=e$ (That's because $(abc)^2=a^2b^2c^2=e$ which means $abc\in \{a,b,c,e\}$. If $abc=a$ then $b=c^{-1}=c$ which is false, same goes for $abc=b,abc=c$.)   I haven't managed to find a reason why the claim must hold for $5, 7, ...$ (i.e. All other odd integers). Does anyone have an idea?  Thanks in advance. Please excuse my English.  edit: Not a canonical answer but a simple one (if such even exists).
Why and how do "bigger pixels" make a better picture? This is going off of today's iPhone 5S announcement, when Phil Schiller said that "bigger pixels" is the key to a better picture.      What does this mean? Is it true or just a marketing ploy? Error: invalid environment block. Press any key to continue...   The error message appears immediately after the boot screen on purple background. Ubuntu 12.04.1 LTS. Increasing the size of tyres on a hybrid We are planning on doing some mountain biking in a few weeks time and we'd like to get some wider tyres for my wife's hybrid. I know there are similar questions and answers out there on this topic, but I need some specific help to make sure I buy the right tyres.  The bike is a Ladies Scott Sportster 50 2015, the best website I could find with a fairly accurate spec is     It's currently got Kenda K-180 700 x 38c / 30TPI tyres on it, which are fairly good all rounders and capable of running on clean forest paths but I don't think they would handle anything heavier.  It's got disk brakes so there isn't a clearance issue there, but I have noticed the frame is generally narrower than true MTBs, here are some photos i took of it.    Has anyone put larger, knobblier tyres on this kind of hybrid? whats the largest I can get away with? - I'm also slightly confused by all the tyre sizing, I'm sure someone told me this is a 27.5" tyre, but 700 seems to mean 29" on a lot of websites.  Any advice is very much appreciated, Thanks. Rows to column conversion of file Suppose I have a file:  File1:   PAPER  TEAM  MANISH NISHA GARIMA JYOUTI ........etc    File2 I want:  PAPER     TEAM MANISH NISHA GARIMA     JYOUTI   Rows to column conversion of File1. Multiple (diamond) inheritance compiles without "virtual", but doesn't with Let $W^{1,p}(U)$ be the Sobolev space. Suppose that $U$ is connected bounded domain in $\mathbb{R}^n$ and $u \in W^{1,p}(U)$ satisfies $Du=0$ a.e. in $U$. How can I prove that $u$ is constant a.e. in $U$? I have written below Wrapper class.  global class BI_VA_AutomationWrapper {      global class BI_VA_AutomationRequest {      webservice String userName;     webservice String emailId;     webservice String buinessUnit;     webservice String orgName;     webservice String country;     webservice String environment;     webservice String myShopNumber;     webservice String BidsId;//for testing     webservice String Type; //for testing  }  global class BI_VA_AutomationResponse {      webservice String responseStatus;     webservice String myShopNumber;     //webservice list&lt;Error&gt; errorMessages;     webservice String  errorMessages;//for testing   } global class BI_VA_Error {      webservice String type;     webservice String message;    }}   And test class is,  @isTest public class BI_VA_AutomationWrapperTestClass  { public static testMethod void unitTest(){      BI_VA_AutomationWrapper.BI_VA_AutomationRequest wrap= new BI_VA_AutomationWrapper.BI_VA_AutomationRequest();        wrap.userName='Test@test.com';     wrap.emailId='Test@test11.com';     wrap.buinessUnit='AH';     wrap.orgName='ORD';     wrap.country='US';     wrap.environment='QA';     wrap.myShopNumber='123456789';     wrap.BidsId='123';     wrap.Type='ResetPassword';              BI_VA_AutomationWrapper.BI_VA_AutomationResponse res= new BI_VA_AutomationWrapper.BI_VA_AutomationResponse();     res.responseStatus = 'Success';     res.myShopNumber = '987654321';     res.errorMessages = 'Test error';      BI_VA_AutomationWrapper.BI_VA_Error error = new BI_VA_AutomationWrapper.BI_VA_Error();     error.type = 'Test';     error.message = 'Test';  }}   Code coverage is 0% only. Is taking the CUA (certified Usability analyst) certification worth it? I see a lot of people taking this  nowadays and I wonder if its actually worth it and how is it received by the industry and does it really help in bringing you up to speed with UX and usability best practices. Can anyone who has taken the exam before provide inputs on its benefits and also its cons if any.  I did see the question  and David's answer did mention that he found it useful but I would like to get more context in what sense people might find it useful and how it could potentially supplement or enhance a career. Scorched3d worked then stopped. Unistalled and reinstalled did not help. Running Ubuntu 16.04.04LTS. I think it was after installing new release.  I think the problem is the Ubuntu 16.04.4LTS. The newer version has no Scorched3d in the Ubuntu Software Center. I have 3 systems. After updating to new version. Scorched3D will not install and work after updating.  ron@RonsComputer:~$ sudo objdump -p $(type -p scorched3d) | grep NEEDED  NEEDED libwx_gtk2u_adv-3.0.so.0  NEEDED libwx_gtk2u_core-3.0.so.0 NEEDED libwx_baseu-3.0.so.0 NEEDED libfreetype.so.6 NEEDED libz.so.1 NEEDED libSDL-1.2.so.0 NEEDED libSDL_net-1.2.so.0 NEEDED libexpat.so.1  NEEDED libstdc++.so.6 NEEDED libm.so.6 NEEDED libgcc_s.so.1 NEEDED libpthread.so.0 NEEDED libc.so.6   ron@ron-desktop ~ $ objdump -p $(type -p scorched3d) | grep NEEDED | awk '{print $2}' | xargs -n 1 dpkg -S  libwxgtk3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_gtk2u_adv-3.0.so.0.2.0 libwxgtk3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_gtk2u_adv-3.0.so.0 libwxgtk3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_gtk2u_core-3.0.so.0.2.0 libwxgtk3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_gtk2u_core-3.0.so.0 libwxbase3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_baseu-3.0.so.0.2.0 libwxbase3.0-0v5:amd64: /usr/lib/x86_64-linux-gnu/libwx_baseu-3.0.so.0 libfreetype6:amd64: /usr/lib/x86_64-linux-gnu/libfreetype.so.6.12.1 libfreetype6:i386: /usr/lib/i386-linux-gnu/libfreetype.so.6 libfreetype6:amd64: /usr/lib/x86_64-linux-gnu/libfreetype.so.6 libfreetype6:i386: /usr/lib/i386-linux-gnu/libfreetype.so.6.12.1 zlib1g:i386: /lib/i386-linux-gnu/libz.so.1.2.8 zlib1g:amd64: /lib/x86_64-linux-gnu/libz.so.1 zlib1g:i386: /lib/i386-linux-gnu/libz.so.1 zlib1g:amd64: /lib/x86_64-linux-gnu/libz.so.1.2.8 libsdl1.2debian:amd64: /usr/lib/x86_64-linux-gnu/libSDL-1.2.so.0.11.4 libsdl1.2debian:i386: /usr/lib/i386-linux-gnu/libSDL-1.2.so.0 libsdl1.2debian:i386: /usr/lib/i386-linux-gnu/libSDL-1.2.so.0.11.4 libsdl1.2debian:amd64: /usr/lib/x86_64-linux-gnu/libSDL-1.2.so.0 libsdl-net1.2:amd64: /usr/lib/x86_64-linux-gnu/libSDL_net-1.2.so.0 libsdl-net1.2:i386: /usr/lib/i386-linux-gnu/libSDL_net-1.2.so.0 libsdl-net1.2:amd64: /usr/lib/x86_64-linux-gnu/libSDL_net-1.2.so.0.8.0 libsdl-net1.2:i386: /usr/lib/i386-linux-gnu/libSDL_net-1.2.so.0.8.0 libexpat1:i386: /lib/i386-linux-gnu/libexpat.so.1.6.0 libexpat1:amd64: /lib/x86_64-linux-gnu/libexpat.so.1 libexpat1:amd64: /lib/x86_64-linux-gnu/libexpat.so.1.6.0 libexpat1:i386: /lib/i386-linux-gnu/libexpat.so.1 libstdc++6:amd64: /usr/share/gdb/auto-load/usr/lib/x86_64-linux-gnu/libstdc++.so.6.0.21-gdb.py libstdc++6:i386: /usr/lib/i386-linux-gnu/libstdc++.so.6 libstdc++6:amd64: /usr/lib/x86_64-linux-gnu/libstdc++.so.6 libstdc++6:amd64: /usr/lib/x86_64-linux-gnu/libstdc++.so.6.0.21 libstdc++6:i386: /usr/lib/i386-linux-gnu/libstdc++.so.6.0.21 libstdc++6:i386: /usr/share/gdb/auto-load/usr/lib/i386-linux-gnu/libstdc++.so.6.0.21-gdb.py libc6:amd64: /lib/x86_64-linux-gnu/libm.so.6 libc6:i386: /lib/i386-linux-gnu/libm.so.6 libgcc1:amd64: /lib/x86_64-linux-gnu/libgcc_s.so.1 libgcc1:i386: /lib/i386-linux-gnu/libgcc_s.so.1 libc6:i386: /lib/i386-linux-gnu/libpthread.so.0 libc6:amd64: /lib/x86_64-linux-gnu/libpthread.so.0 libc6:i386: /lib/i386-linux-gnu/libc.so.6 libc6:amd64: /lib/x86_64-linux-gnu/libc.so.6 ron@ron-desktop ~ $ Smooth option sometimes produces incorrect arrow tips in PGFplots It seems that the smooth option applied to lines sometimes effects the arrow tips and produces incorrect results.  No problem with vertical lines, but any other seems to have a problem. Is this a known bug, or is there a reason why the smooth should not be applied to straight lines?  Updated: to show problem does NOT occur with TikZ, and problem also occurs with curves if smooth option is applied.  \documentclass{article} \usepackage{pgfplots} \begin{document}  \tikzstyle{MyStyle}      =[-&gt;, black, ultra thick,       ] \tikzstyle{MyStyleSmooth}=[-&gt;, black, ultra thick, smooth]  \begin{tikzpicture} \begin{axis}     % 'smooth' option seems to have no effect on lines '1' and '2'     \addplot [MyStyle      , green ] coordinates{(0,0) (0,1)} node [above] {1};     \addplot [MyStyleSmooth, blue  ] coordinates{(1,0) (1,1)} node [below] {2};     %     % But, can't seem to have the 'smooth' on lines, '4', '5' and '6'     \addplot [MyStyle      , orange] coordinates{(2,0)   (3,0)   } node [right] {3};     \addplot [MyStyleSmooth, red   ] coordinates{(4,0)   (2,0.5) } node [below] {4};     \addplot [MyStyleSmooth, red   ] coordinates{(2,1)   (3,1)   } node [left ] {5};     \addplot [MyStyleSmooth, red   ] coordinates{(3,0.8) (2,0.8) } node [right] {6};      % Similar problem with curves: "non-smooth" is ok, "smooth" is not     \addplot[MyStyle,       mark=none, domain=0:1,samples=50, blue] (x,x*x+1);     \addplot[MyStyleSmooth, mark=none, domain=0:1,samples=50, red ] (x,x*x+2);  \end{axis} \end{tikzpicture}  \begin{tikzpicture} % These are all fine     \draw [MyStyle      , green ] (0,0)   -- (0,1)    node [above] {1};     \draw [MyStyleSmooth, blue  ] (1,0)   -- (1,1)    node [above] {2};     %     \draw [MyStyle      , orange] (2,0)   -- (3,0)    node [right] {3};     \draw [MyStyleSmooth, red   ] (4,0)   -- (2,0.5)  node [below] {4};     \draw [MyStyleSmooth, red   ] (2,1)   -- (3,1)    node [right] {5};     \draw [MyStyleSmooth, red   ] (3,0.8) -- (2,0.8)  node [left ] {6}; \end{tikzpicture} \end{document} Return single column from a multi-dimensional array Prove that $\overline {A^\complement} = (A°)^\complement $   To show that $\overline {A^\complement} \subseteq  (A°)^\complement$   We first show $A^\complement \subseteq  (A°)^\complement ...(*) $  We know that since $ A°\subseteq A \qquad (*)$ is automatically follows .  The part I didn't understand is the following:     Then we conclude that since $ (A°)^\complement$ is closed    $\therefore \overline {A^\complement} \subseteq  (A°)^\complement$ Verify that $4(29!)+5!$ is divisible by $31$. What does  signify when using Vim LatexSuite? I have a LaTeX editing environment that does some assistive stuff such as auto-indenting, closing parentheses, etc.  However, one of the things that it does is unknown to me, and is difficult to search for. If I type an empty set of braces, it replaces them with  {}&lt;++&gt;   If I type two empty sets, for example when I'm setting up a fraction \frac{}{} what I actually get is  \frac\left\{ } \right\}&lt;++&gt;}&lt;++&gt;   Is this a problem? What does &lt;++&gt; do?  Edit: This is using Search all tags with no tag wiki Is there a way I can search the Tags for tags with no tag wiki created yet? Prove: If a sequence converges, then every subsequence converges to the same limit. What is the fastest way to create a 90° joint? I often want to create joints like this:    Since I end up making these more than a little, I'd like a fast way to do it (i.e. on the order of &lt; 10 seconds)  I've found a couple tools which come close to being the ideal solution, but I've had some problems with them as listed below:   When using the spin tool I usually end up having to fiddle with the location for a while in order to to get a natural curve:    Are there any techniques to expedite this process of fiddling? The bridge tool is nice because I can precisely specify where both ends of the joint will be, however I often end up with subtly uneven joints, probably due to my initial mesh being not quite square.    Is there a fast way to correctly line up the initial edge loops in a way that will result in an even joint? I am looking for an example such that in a pre-Hilbert space $H$ we have for a subspace $U$ that  (i) $\bar{U} \oplus U^\perp \neq H$  (ii) $ \bar{U} \neq U^{\perp \perp}$  Since finite and closed subspaces will probably not do it, I am not good at inventing nice examples. Does anybody here have a few at hand? Detect which TeX engine is used How can I detect inside a TeX file which engine (pdftex/xetex/luatex) is used to compile the file? Is there a LaTeX package that provides this functionality? It shows no question owner, not even a simple name (without link).   What happened?