Show that there exists a non-trivial automorphism on $G$.
$|G|>2$ implies $G$ has non trivial automorphism
Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$ Skins that cannot be in mystery gifts? Yesterday I had a discussion with some of my friends while playing league. It was about the contents of the mystery gifts and while I said there are restrictions, they were convinced that you can get absolutely every skin from a gift (including victorious, and really old ones).  Now while I'm pretty sure that there are restrictions (Triumphant ryze for example), I'm not really sure on which skins they affect.   What about the victorious, riot, legacy, pax and completely removed skins (Rusty BC, Black Ali, Urf WW)? Can you get those with a mistery gift or is it impossible to get those right now? Finish an activity after a time period I'm just going to throw this out there, but Google indexes Stack Overflow really quickly Misplaced navigation bar and question title Embedding ordinals in $\mathbb{Q}$ All countable ordinals are embeddable in $\mathbb{Q}$.  For "small" countable ordinals, it is simple to do this explicitly.  $\omega$ is trivial, $\omega+1$ can be e.g. done as $\{\frac{n}{n+1}:n\in \mathbb{N}\} \cup \{1\}$.   $\omega*2$ can be done as $\{\frac{n}{n+1}:n\in \mathbb{N}\} \cup \{1+\frac{n}{n+1}:n\in \mathbb{N}\}$, and $\omega*n$ as $\bigcup_{i\le n} \{i+\frac{n}{n+1}:n\in \mathbb{N}\}$  That immediately also gives $\omega^2$ as  $\bigcup_{i \in \mathbb{N}} \{i+\frac{n}{n+1}:n\in \mathbb{N}\}$  And going further is still relatively easy: We can biject the above embedding of $\omega^2$ onto a single interval $(0,1)$, e.g. through $f(x)=1-\frac{1}{x+1}$ since this is an order preserving bijection from $\mathbb{Q}^+$ to $(0,1)$, allowing us to get $\omega^2+n$, $\omega^2*n$ etc. And by iterating that process, we can get any ordinal below $\omega^\omega$.   But this sort of embedding fails at $\omega^\omega$ as the iteration of $f$ doesn't seem compatible with taking the infinite union - figuratively speaking, we'd squish things to $0$. So what would an explicit embedding of $\omega^\omega$ look like? And the question, then, is if it is possible to give an explicit embedding of $\epsilon_0$? Is the fact that Peano arithmetic cannot prove well-foundedness of $\epsilon_0$ an indication that it cannot be embedded by such elementary functions and their iterations?  And what about even bigger countable ordinals such as the Veblen ordinals, Feferman-Schütte, Bachmann-Howard? From its definition, even if the above all are possible, I assume that the definitive bound of where one can define an effective procedure for the embedding must be Church–Kleene - is that a correct conclusion?  Lastly, does the situation change when we use $\mathbb{R}$ instead of $\mathbb{Q}$, which would allow us to use, perhaps, more easily understood yet inherently complex functions?  Edited because $\omega^2 \neq \omega^\omega$ using a:hover to change the color of an image Beamer - create subsubsection page There exists an injection from $X$ to $Y$ if and only if there exists a surjection from $Y$ to $X$. Theorem. Let $X$ and $Y$ be sets with $X$ nonempty. Then (P) there exists an injection $f:X\rightarrow Y$ if and only if (Q) there exists a surjection $g:Y\rightarrow X$.  For the P $\implies$ Q part, I know you can get a surjection $Y\to X$ by mapping $y$ to $x$ if $y=f(x)$ for some $x\in X$ and mapping $y$ to some arbitrary $\alpha\in X$ if $y\in Y\setminus f(X)$. But I don't know about the Q $\implies$ P part.  Could someone give an elementary proof of the theorem? What is the maximum number of people that will join a settlement? I've got two settlements on the go that I'm paying attention to - Sanctuary and the Starlight Drive In. Both of these settlements have radio towers and beds and all resources in the green however they appear to have stopped growing at 10 people in each settlement even though happiness is above 80 and I've got 15+ beds at each location.  What is the maximum number of people that will join a settlement? Problem with IMPORTRANGE function in Google Sheets I’m trying to get IMPORTRANGE to work for me with no success. I made two simple spreadsheets:  and . In the destination sheet I’m using the import range function:  =importrange(“1-5F1_zz0ZpGj7SfeQhewsCe0CsrA0eyPbjfmUrLBsyk”,”Sheet1!a1:a1”)   I tried already:   Replacing the , with ; Referencing just the key as well as the full URL in the first argument Switching from “ to ‘ With the sheet name and without it   Not sure what else to do.  Any ideas? jQuery ajax load() java script not executing? How to retain HTML5 Attributes on Markup I have a page with certain html markup and element attributes as shown in the snippit below:  &lt;ul class="accordion" data-accordion=""&gt;  When I edit the page as a user with the WP Administrator Role the &lt;ul&gt; remains intact upon page update. Perfect.  Yet when a user with WP Editor Role edits the page, the ul tag saves as follows:  &lt;ul class="accordion"&gt;  Notice the data-accordion attribute is removed.  How can I get WordPress to retain markup tag attributes for users with the Editor role? PSTricks error "Undefined control sequence  \c@lor@to@ps" I have a problem with this file tex:  \begin{document} \begin{figure} \begin{center} \begin{pspicture}(0,0)(1,3.8) \psline{-&lt;}(0,0)(0,.2) \psline{-}(0,.2)(0,.4) \psline{*-&gt;}(0,.4)(0,.9) \psline{-*}(0,.9)(0,1.4) \psline{-&lt;}(0,1.4)(0,2.4) \psline{-}(0,2.4)(0,3.4) \psline{*-}(0,3.4)(0,3.6) \psline{&gt;-}(0,3.6)(0,3.8) \rput(1,.4){$y=0$}%testo \rput(1,1.4){$y=1$} \rput(1,3.4){$y=3$} \end{pspicture} \end{center} \caption{} \end{figure} \end{document}   but I don't understand where is the error. How do I drill a 2⅛ inch door knob hole over an existing 1½ inch hole? I need to replace the door knob for this interior door:    The door knobs I've found at the stores require a 2⅛ inch hole.  Mine is only 1½:    Did door knobs used to be smaller?  Is there some place I can find a replacement door knob that will fit my door?  If I need to modify the door to fit a modern door knob, will the new hole completely overlap the old? and can I reuse the latch hole? Proving the Cauchy-Schwarz inequality by induction Should I always use a comma after "e.g." or "i.e."? It seems that "e.g." is always followed by a comma but "i.e." is not. Why is that? Hill Cipher known plaintext attack I know a plaintext - ciphertext couple of length 6 for a hill cipher where its key is a [3x3] matrix.  Based on what I've read and learned, to attack and crack keys of [n x n], if we know a plaintext - ciphertext duo of length $n^2$ then we have our set of n equations with n variables, and this is generally solvable. However in my case there is only length of $6$ instead of $n^2 = 9$ .   My question is, how will I solve this problem and find the key? Or, since this is an obvious homework question, what should be my way of thinking? I cannot think of anything else but matrix multiplications and inverses but they do not help me at all. The staircase paradox, or why $\pi\ne4$ What is wrong with this proof?    Is $\pi=4?$ Can anyone explain why a semi-simple finite dimensional Lie algebra $\mathfrak{g}$ has to be perfect ?   The natural way to prove something like that would be to look to the algebra generated by the Lie brackets, which when $\mathfrak{g}$ is not perfect would be expected to be solvable. But it doesn't seem to work.