I'm stuck in proving that the order of an element divides the order of the group.  I've already proved Lagrange's theorem that for $H&lt;G, |H|||G|$   And I know that for any $a\epsilon G, |a|=|&lt;a&gt;|$  The problem is that though I know this for a fact, I need to prove that the order of the element equals the order of the cyclic subgroup it generates but how do i prove this result to finally conclude that the order of a divides G using lagrange?  Please help.
Proving this can be done as follows: consider a finite group G and elements $g_i \in G$ for some integer $i$. Now consider $\langle g_i \rangle = \{g_i^n: n\geq 0\}$, a generator. It can be proved that $\langle g_i \rangle \leq G$ and that the order of $g_i$ is equal to the order of $\langle g_i \rangle$, so $|\langle g_i \rangle| \leq |G|$.  We can now use Lagrange's theorem which states that if $H \leq G$ then $|H|$ divides $|G|$ and we're done. But...  Is there a simpler way to prove this fact?
Now this might be a very stupid question for many readers, but I am getting really confused.   I was reading about electricity and current, and I read this:  Suppose we connect a wire to the ends of a battery, and a bulb in the circuit, current flows and lights up the bulb. The battery is responsible for doing work on the positive charges to move them from negative terminal to positive terminal, that is moving against the field. Now, these energised positive charges move through the external circuit, and within the circuit they give some of their energy to light the bulb. I understood this very well.  Now, reading further, it was written that "Electrons are actually responsible for current flow". It was written that there is concentration of energised negative charges on the negative terminal, and these electrons then move through the external circuit from negative to positive terminal. This is really confusing.     So my question: What actually flows? The negative charge or the   positive charge?   Further reading, I also read that in semi-conductors, fluorescent light bulbs etc., both charges move through the circuit.     Now, one more doubt: If both charges move through the circuit in this   case, shouldn't the charges stop in the middle of the wire, because   opposite charges will attract each other, hence stopping their   movement? How To Program A Bash Script That Presses A Certain Combination Of Keys? Please, i need someone to help me in making a very simple bash script. The job of this script is just to press a certain combination of keys on the keyboard.  My objective is to create a script that would turn off the Kubuntu desktop effects completely. The key combination for that is: "Left-ALT + Left-Shift + F12". I am aware that there is a widget that might do this job, but i am trying to learn how to accomplish it with a bash script.  I have tried to do an online search but i guess i was choosing the wrong set of keywords. I did not manage to reach a tutorial or a reference that would mention this task.  Thanks in advance for your help!  P.S: Any suggestions, references, notes, comments or tutorials would also be greatly be appreciated. Do I end up richer paying down a mortgage or contributing to an IRA? I have $10,000 extra cash that I can either use to pay down my 5% mortgage, or contribute to my IRA.  I'm 30 years old.  I hope to pay off the mortgage in about 5 years.  I'm stymied, because:   Paying $10,000 into my mortgage earns me $500 a year (in interest savings) for the life of the mortgage -- hopefully 5 years. Paying $10,000 into my IRA earns me roughly a $3000 lowering of my tax bill now (plus CD ROI of about 1% at retirement time.)   So I can get $2500 ($500 a year over 5 years), after which I will have paid my mortgage off with other money; or I could get $3000 in cash right now plus a bit of extra money when I retire.  Even ignoring the future benefits of the IRA contribution, it feels like I should put the money into my IRA: it's an immediate 30% return on investment, as opposed to a 5% return for 5 years.  But if I always make that decision, I won't be paying off my mortgage early, and that $500 will stretch out over 30 years, costing me $15,000!  Which actually leaves me with more money in the end?  What basic principle of math am I missing?  Update:  The principle of math I was missing is that, assuming I'm taxed at the same rate when I retire as I am taxed now, I end up with exactly as much money as if I had invested the $10k and not received a $3k tax reduction.  (Of course, my income level and tax laws will have changed by then, but let's ignore that.)  So the question simplifies to, "Should I put $10k toward eliminating a 5% debt or toward a 1% investment?" in which case I should clearly put it toward the mortgage.  I was confused by the government incentive into thinking that retirement investment was a sweeter deal than it actually is.  This is probably their goal, and probably a noble one :) Is there a Raster Equivalent of the Split tool? Are Tolkien Elves taller or shorter than Men?  In general fantasy, elves are frequently smaller than humans. But I seem to recall that in Tolkien they aren't.  Based on Is there a "null coalescing" operator in JavaScript? Is there a null coalescing operator in Javascript?  For example, in C#, I can do this:  String someString = null; var whatIWant = someString ?? "Cookies!";   The best approximation I can figure out for Javascript is using the conditional operator:  var someString = null; var whatIWant = someString ? someString : 'Cookies!';   Which is sorta icky IMHO. Can I do better? What are the light blue lines on my imported objects Normal subgroups of infinite symmetric group I recently took a course on group theory, which mentioned that the following proposition is equivalent to the continuum hypothesis: "The infinite symmetric group (i.e. the group of permutations on the set $\mathbb{N}$) has exactly 4 normal subgroups." Does anyone have any references or explanation for this? I saw a dog.       I have seen a dog.   What are the differences between them? Did these events happen on the same day? When a ball is tossed straight up, does it experience momentary equilibrium at top of its path? This question has been asked many times all over the Internet and answers can be found on places such as yahoo and ask.com, but I'm not satisfied with those answers and I don't trust the validity of those places.  This seems like a more appropriate place to ask.  So, here's my train of thought and I would like to ask you if you think I'm right or wrong.  Even though most of the answers point to the fact that the ball does not achieve equilibrium because the force of gravity is constantly acting upon it, thus causing an acceleration, I still think there's a moment of equilibrium and here's why:  When the ball is traveling up, it's accelerating towards the ground and eventually reaches a point at which its speed reaches 0.  At this moment, isn't the reason the speed is 0 is due to the fact that net force is 0?  The force that made the ball travel in the upward direction was canceled out by the gravitational force.  So, the sum of forces at that brief moment is equal to 0, otherwise the ball would be moving.  I don't think gravity is the only force acting on the ball at that moment, since the throwing force was acting on it as well, until both canceled each other out for that brief moment.  Am I right or wrong? Adding multiple CKEditor instances in jquery Combining Multiple Lists How does accepting an answer work? What does it mean to accept an answer? How do I accept an answer, and what are the rules? Can I accept my own self-answer? What are the reputation benefits? Which answer should I accept? Should I upvote the accepted answer? What should I do if none of the answers are suitable?      For more information, see "" in the . Is there consensus in the field of statistics that one book is the absolute best source and completely covering every aspect of GLM - detailing everything from estimation to inference? I would appreciate if someone could help me write the mathematical equation for the seasonal ARIMA (2,1,0) x (0,2,2) period 12. I'm a little confused with how to go about this. I would prefer an equation involving $Y_t , e_t, θ$ and $Θ$. How do you add a timed delay to a C++ program? I am trying to add a timed delay in a C++ program, and was wondering if anyone has any suggestions on what I can try or information I can look at?  I wish I had more details on how I am implementing this timed delay, but until I have more information on how to add a timed delay I am not sure on how I should even attempt to implement this. Are Ubuntu packages security audited? I am trying to find out whether $A = \begin{bmatrix}     3 &amp; 1 \\     0 &amp; 2 \\ \end{bmatrix}$ and $B = \begin{bmatrix}     4 &amp; 1 \\     -2 &amp; 1 \\ \end{bmatrix}$ are similar. Both matrices have the same trace, same rank, same determinant and the same eigenvalues. So I think they are similar. The eigenvalues are $3, 2$. Now I'm not sure how to find if they are indeed similar.   I am trying to find if there is a basis for $\mathbb{R^2}$ that contains only eigenvectors but when trying to solve $(3 I - A)v = 0$ there is no solution... What should I do from here? I want to subtract the current frame from a previous frame, I have a image sequence and I want blender to subtract the current frame from the frame with an offset of -4. If I am on frame 5 so blender will subtract and use frame 1. An acquaintance recently posted an image to FB from his recent trip to the JFK Historic Site asking for an explanation of a visible artifact he states was not visible to the naked eye but was visible through the EVF of his camera and in the final image.    The artifact is a series of bright, nearly-vertical bars and a longer, almost horizontal bar superimposed over the lower part of the door frame and the radiator.  I found a very similar artifact in another image of the same room from a blogger which isn't as compressed:    In both cases the artifact seems to be a vertically inverted image of the post and rail outside the saturated window.  From looking at other images of the room available online I know there is spherical glass ornament topping the table lamp whose shape is visible at the left of the photo.  One response to the OP's question is as follows:     RK: It is lens flare. So, even on an iPhone, the camera lens is made   up of multiple lenses. Each designed to focus the image and make sure   it isn't distorted. Remember a lens bends light.  Sometimes, based on   angle and intensity of the incoming light, some stray light bounces   back and forth while traveling through the various elements that make   up what we think of as a camera lens (some of mine have 16 elements   inside a single lens.) Even the best glass is capable of reflecting   some light instead of passing it through the lens elements to the   sensor.  This is what you are seeing. The railing is overexposed, and   some stray light is bouncing around a bit inside the lens.   My response:     DJ: I don't think it's lens flare or ghosting. Those usually form an   image of starbursts, rings, or circles in a row across the image or in   the shape of the aperture; or a veiling glare or haze. This is forming   an image of the light outside minus objects in its path. It is more   likely a direct reflection/projection off a mirror or the glass in a   picture frame elsewhere in the room onto the opposite wall and the   radiator. There are two other clues which seem to confirm this. First,   the image has a gradiated appearance, it is less distinct in the   relatively bright area on the door frame and brighter in the   relatively darker areas behind and on the radiator. You would expect   lens flare to be more evenly specular. Second, the image representing   the light from below the horizontal rail changes angle on the   different faces of the door frame. Unlike a projection, lens flare is   an image created within the lens, the physical shape of the objects it   superimposes do not change its shape.      OP: If that were the case, would I have been able to see said   reflection in the room without the camera? There was no image I could   see with the naked eye. I notice the image in my phone before I took   the pic and did a double, triple and quadruple take before I took it   so I'm sure.      DJ: It would have been visible at the moment the image was taken, but   it may not have appeared nearly as bright and distinct because your   eyes were accommodating for the bright light coming in the window. The   projection could have easily been obscured by you or anyone around you   changing position or moving through the path of the projected   reflection. Also, it's been caught by other people... (see Image 2)   Notice that, in the linked example, the projection is again distorted   by the surface it falls on.   Anyone else want to chime in with an explanation?