If $G$ is a finite group, then $G$ is isomorphic to some subgroup of $S_n$.  I was able to prove this theorem by using the fist part of Cayley's theorem.   But I can't prove that $S_n$ is isomorphic to a group of permutations. How can I show that?
This is from Fraleigh's First Course in Abstract Algebra (page 82, Theorem 8.16) and I keep having hard time understanding its proof. I understand only until they mention the map $\lambda_x (g) = xg$. Can someone explain this proof step by step? Thank you so much! Here is the proof:  Let $G$ be a group. We show that $G$ is isomorphic to a subgroup of $S_G$. Define a one-to-one function $\phi: G \to S_G$ such that $\phi(xy)=\phi(x) \phi(y)$ for all $x,y \in G$. For $x \in G$, let $\lambda_x: G \to G$ be defined by $\lambda_x (g) = xg$ for all $g \in G$. The equation $\lambda_x (x^{-1} c) = x(x^{-1} c) = c$ for all $c \in G$ shows that $\lambda_x$ maps $G$ onto $G$. If $\lambda_x (a) = \lambda_x (b)$, then $xa=xb$ so $a=b$ by cancellation. Thus $\lambda_x$ is also one to one, and is a permutation of $G$. We now define $\phi: G \to S_G$ by defining $\phi(x) = \lambda_x$ for all $x \in G$. To show that $\phi$ is one to one, suppose that $\phi(x) = \phi(y)$. Then $\lambda_x = \lambda_y$ as functions mapping $G$ into $G$. In particular $\lambda_x (e) = \lambda_y (e)$, so $xe=ye$ and $x=y$. Thus $\phi$ is one to one. It only remains to show that $\phi(xy) = \phi(x) \phi(y)$, that is, that $\phi_{xy} = \lambda_x \lambda_y$. Now for any $g \in G$, we have $\lambda_{xy} (g) = (xy)g$. Permutation multiplication is function composition, so $(\lambda_x \lambda_y)(g) = \lambda_x (\lambda_y (g)) = \lambda_x (yg) = x(yg)$. Thus by associativity, $\lambda_{xy} = \lambda_x \lambda_y$.
multiple definition error including c++ header file with inline code from multiple sources I have a c++ header file containing a class. I want to use this class in several projects, bu I don't want to create a separate library for it, so I'm putting both methods declarations and definitions in the header file:     // example.h #ifndef EXAMPLE_H_ #define EXAMPLE_H_ namespace test_ns{  class TestClass{ public:     void testMethod(); };  void TestClass::testMethod(){     // some code here... }  } // end namespace test_ns #endif   If inside the same project I include this header from more than one cpp file, I get an error saying "multiple definition of test_ns::TestClass::testMethod()", while if I put the method definition inside the class body this does not happen:  // example.h #ifndef EXAMPLE_H_ #define EXAMPLE_H_ namespace test_ns{  class TestClass{ public:     void testMethod(){         // some code here...     } };  } // end namespace test_ns #endif   Since the class is defined inside a namespace, shouldn't the two forms be equivalent? Why is the method considered to be defined twice in the first case? Convolution operator has norm 1 with respect to $L^2$ norm Let $C_c(\mathbb{R})$ be the following:  $$C_c = \{ f \in C(\mathbb{R}) \mid \exists \text{ } T &gt; 0 \text{ s.t. } f(t) = 0  \text{ for } |t| \geq T\}$$  Let $T_n \in L(C_c(\mathbb{R}))$ be a linear operator such that:  $$T_n u = \delta_n *u, \forall u \in C_c(\mathbb{R}),$$  where  $$ \delta_n(t)= \begin{cases}        n^2(t+1/n) &amp; -1/n \leq t \leq 0 \\       -n^2(t-1/n) &amp; 0 &lt; t \leq 1/n \\       0 &amp; \text{elsewhere}     \end{cases} $$  I have to prove that with respect to the 2-norm, the operator $T_n$ has $\|T_n\| = 1$ and I have succeeded in proving that $\|T_nu\|_2 \leq \|u\|_2, \forall u \in C_c(\mathbb{R}).$  Now, I remained  to prove that $\exists \text{ } u \in C_c(\mathbb{R}) \text{ s.t. } \|T_nu\|_2 = \|u\|_2$, but I really don't get how $u$ should be shaped in order to satisfy it. Hi there! I'm looking for a help drawing this diagram. Any ideas? Which is correct: "he don't" or "he doesn't"? Which one is correct in a sentence?        He don't    He doesn't      I guess "he doesn't" should be correct because he is third person singular but I've seen some people using do with he. Which one is correct? I need a formula for conditional formatting to make an Excel 2010 spreadsheet alternate highlighting when a single column changes.  This question, "", is exactly what I want to do, but the suggested solution didn't work for me--it just highlights the first row of each matching value.  I know how to use conditional formatting, but for the life of me I can't figure out or find any pointers on the net so far to make this work.  Ideas?  Here is a link to a picture of how I want my spreadsheet to look when I'm done.  Basically I want every other Disp Number value row to be highlighted.  Seems like it would be a common thing to want to do, and I've seen this asked for in various places, but people struggle with making it work. Display error message if same record exist with combination of some fields? Does the diameter of the aperture increase with increases in the focal length with constant F-number(F-stops)? What I have learned from the definition of F-number is that;  The F-number (sometimes also called focal ratio, f-ratio, f-stop, or relative aperture) of a lens is the ratio of its focal length divided by the diameter of the aperture. And F-numbers are expressed as f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32 etc. Hence F-number (F-stop) depends on two variables; that are diameter of aperture and focal length of the lens.  Aperture diameter/size is one of the component which governs the amount of light that enters to the camera. As far as I known, at different F-stops values (f/1, f/1.4, f/2, f/2.8, f/4, f/5.6, f/8, f/11, f/16, f/22, f/32) the amount of light allowed to enter the camera is fixed and are with either 1/2 or 1/3 stop increment.  This concept is very easy to understand, in case of prime lenses (with a  fixed focal length and variable aperture), where F-stop is inversely proportional to the aperture diameter. For lower F-number (F-stop) the aperture would be bigger allowing more light and for higher Hence F-number (F-stop) the aperture would be smaller allowing less light in to the camera.  F-numbers (F-stops) have been evolved to allow a definite/fixed amount of light in to the camera (As far as I known). In case of zoom lens (with variable aperture and variable focal length), I am not able to understand the what would be the relationship between F-stop and the diameter/area of aperture at different focal length.  I have two related questions  1 . In case of zoom lens,  for example, Canon EF-S 18-135mm f/3.5-5.6  (with variable aperture and variable focal length). At any particular F-number(F-stop), for example at f/5.6 , does the diameter of the aperture increase with increases in the focal length? I mean,  Is the diameter of the aperture at  18mm and f/5.6 would be smaller than that of at 135 mm and f/5.6?   In the above case (i.e. Canon EF-S 18-135mm f/3.5-5.6 ), What is amount of light that would enter the camera at 18mm and f/5.6 compared to that of at 135 mm and f/5.6? Will the amount of lights be same or different? how do i change the color of typeface on photoshop? Some Data Failed to Reconstruct, Motion Tracking I added tracks and stuff, but then when I tried to solve motion track, there was a error that said:     Some Data Failed to Reconstruct   and it also said to look into the console for details, which I did. This is what the console said:    By my understanding, my video is 372 frames long, and Blender is cutting the footage at 318 for some reason. Here is the image of the scene:    Why? Here are links to  and also . Also the next morning I did a camera solve, but then most of the frames went red. Legend in tikzpicture How much willpower do gravely wounded soldiers lose? The text says they suffer a permanent reduction? I am preparing a document in French, using the "babel" package with option "french" and using commands that include colons and semicolons. My problem is, the French spacing before colons and semicolons disappears when the commands are printed. I do not understand why, have not been able to find an answer. Below is a MWE.  \documentclass{article} \usepackage[french]{babel} \newcommand\deuxpoints{Texte: texte} \begin{document} \deuxpoints\ \endgraf Texte: texte \end{document} Rendered image is just grey Maybe a stupid question but how can I tell if I have a static IP address or a dynamic IP address? Running Fedora-13 I understand that unlocking the bootloader will wipe my Android phone, but have been looking around for why.  Seems to be by design, but what is the reasoning for that design?  Is it some security concern, some obscure technical reason, or just for lulz?  I'm looking for something solid to chew on here, something more than because "that's how it is". can I change value of $_SERVER['DOCUMENT_ROOT']? I am new to TeX/LaTeX. How do I install TeX on Windows (7)? It seems that there are many different implementations, so which one is best for my OS? How to convert strings into integers in Python? I have a tuple of tuples from a MySQL query like this:  T1 = (('13', '17', '18', '21', '32'),       ('07', '11', '13', '14', '28'),       ('01', '05', '06', '08', '15', '16'))   I'd like to convert all the string elements into integers and put them back into a list of lists:  T2 = [[13, 17, 18, 21, 32], [7, 11, 13, 14, 28], [1, 5, 6, 8, 15, 16]]   I tried to achieve it with eval but didn't get any decent result yet. Shrink Tempdb log file when reaching a specific size how I can shrink tempdb log file when reaching a specific size?? I'm scheduling job to shrink tempdb but still I face a problem of disk space due to tempdb is full! Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating:  In analytic (Lagrangian) mechanics, the derivation of the Euler-Lagrange equations from the principle of least action assumes that the start and end coordinates at the initial and final times are known. As a consequence, any variation on the physical path must vanish at its boundaries. This conveniently cancels out the contributions of the boundary terms after integration by parts, and setting the requirement for minimal action, we obtain the E.L. equations.  This is all nice and dandy, but our intention is finding the location of a particle at a time in the future, which we do not know a priori; after we derive any equations of motion for a system, we solve them by applying initial values instead of boundary conditions.   How are these two approaches consistent?