Let $p$ is an odd prime and $G$ is a group which has $2p$ elements. Show that there exists at least one element with order $p$.  I tried showing in 2 parts as $G$ is cyclic and not cyclic, but I couldn’t show it for not cyclic groups. I think I must do something different. Could you please help me in the easiest way becasue we have learnt only elemantry theorems and definitions?
I want to show without using Sylow theorem that     Any group of order $2p$ has a subgroup of order $p~$($p$ being a prime)   My attempt: Since $|G|=2p,$ even $\exists~a\neq e\in G$ such that $a^{-1}=a.$ Then $H=\{e,a\}\leq G.$ Let $S$ be the set of all left cosets of $H$ in $G.$ Now $S$ forms a group w.r.t. the composition $g_1Hg_2H=(g_1g_2)H.$ The mapping $\sigma:S\to G:gH\mapsto g$ is an $1-1$ homomorphism. Then $S\simeq\sigma(S)\leq G.$ Thus $G$ has a subgroup of order $p.$  Please tell me whether I'm right!     I'm skecptical as I didn't use that $p$ is prime.
How to edit geometry of multiple layers simultaneously in QGIS 3? I have several line layers (route, pipes, cables) that share the same geometry and would like to edit them simultaneously.   Is there a way to easily snap the geometry of the pipes and cables layers to the route layer so that I can simply change the geometry of the route layer and don't have to repeat all changes on the other two layers as well?  To clarify, I plan a fibreglass route and have three layers. The first one describes the route that I have to take to pass all homes that have to be connected. The second one contains the pipes in which the cables lie and the third layer is the cable layer. So all features on those layers have to have the same geometry. Once I change the pathway of the route layer, I have to apply the same changes to the other two layers to ensure that they are still aligned with the route layer. I'm not sure whether I understand your comments correctly, but I use the basic editing tools in QGIS 3. Set origin to vertex in edit mode? I want to make the origin in edit mode at a vertex, but it keeps setting it to the last clicked-on face :( So now in edit mode the pivot is on the last clicked-on face ;( Largest Mass Diffraction Why isn't it $E \approx 27.642 \times mc^2$? Why is it that many of the most important physical equations don't have ugly numbers (i.e., "arbitrary" irrational factors) to line up both sides?   Why can so many equations be expressed so neatly with small natural numbers while recycling a relatively small set of physical and mathematical constants?  For example, why is mass–energy equivalence describable by the equation $E = mc^2$ and not something like $E \approx 27.642 \times mc^2$?  Why is time dilation describable by something as neat as $t' = \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$ and not something ugly like $t' \approx 672.097 \times 10^{-4} \times \sqrt{\frac{t}{1 - \frac{v^2}{c^2}}}$.  ... and so forth.    I'm not well educated on matters of physics and so I feel a bit sheepish asking this.   Likewise I'm not sure if this is a more philosophical question or one that permits a concrete answer ... or perhaps even the premise of the question itself is flawed ... so I would gratefully consider anything that sheds light on the nature of the question itself as an answer.    EDIT: I just wished to give a little more context as to where I was coming from with this question based on some of the responses:  @Jerry Schirmer comments:      You do have an ugly factor of $2.997458 \times 10^8 m/s$ in front of everything. You just hide the ugliness by calling this number c.   These are not the types of "ugly constants" I'm talking about in that this number is the speed of light. It is not just some constant needed to balance two side of an equation.  @Carl Witthoft answers:      It's all in how you define the units ...   Of course this is true, we could in theory hide all sorts of ugly constants by using different units on the right and the left. But as in the case of $E=mc^2$, I am talking about equations where the units on the left are consistent with the units on the right, irrespective of the units used. As I mentioned on a comment there:     $E=mc^2$  could be defined using units like $m$ in imperial stones ($\textsf{S}$), $c$ in cubits/fortnight ($\textsf{CF}^{−1}$) and $E$ in ... umm ... $\textsf{SC}^2\textsf{F}^{−2}$ ... so long as the units are in the same system, we still don't need a fudge factor.   When the units are consistent in this manner, there's no room for hiding fudge constants. How can I highlight text before a character with listing? Sentences with "no one" Index entries duplicated for captions In document class book, and package caption, creating index with makeidx. Index entries for the main text and for footnotes collate properly; but those for captions create their own entries.  Example: for the tag \index{New York!Custom House!\textit{Cartouche}}, text and footnotes collate properly; but used in a caption, a new entry is created at the third level, and is also out-of-alphabetic-sequence. Note: the top level, and first sub are correct; only the second sub is duplicated.  How to get the caption index entries under the same entry? How do I split a string, breaking at a particular character? I have this string  'john smith~123 Street~Apt 4~New York~NY~12345'   Using JavaScript, what is the fastest way to parse this into  var name = "john smith"; var street= "123 Street"; //etc... Differences among "It feels...", "It looks...", and "It seems..." Can I live in the UK for a long time through repeat visits? I had recently entered the UK without a visa to be with my partner, from 21 July 2016 until 16 December 2016. I hope to return on the 9th of January. We don't meet the income requirement for a Spouse visa. I recently applied to university, and got in, and was intending to return to the UK on a student visa, but my university did not tell me about maintenance fees and my loan wasn't large enough to cover it all, so my visa would have been denied.  Now, I am hoping to attend university in the next semester, and therefore return to the US to apply for that visa. Do you think that if I try to return to the UK without a visa, I will be turned away?  I won't be working or studying during this time, however, I'm afraid that they'll consider it living in the UK illegally; I don't have a criminal record or anything like that. I don't like having the middle mouse button paste, because I often end up with uncompilable code in Eclipse. How can I turn this off (in all programs)? I'm running Fedora. Reprojecting raster from lat/lon to UTM in R? How to add custom C++ keywords to be recognized by Minted? I have defined several structures and types in a C++ project. I have to write some documentation for this code, and I use  to typeset my code.   I would like to be able to add some keywords to the minted database, so minted will also recognize these and give them the appropriate color. (I would like to have them colored in the same way a type, eg. double would be colored.) Can't log in using Firefox in Stack Overflow I want to create a DEB file manually. I would like to just provide a folder which contains data to install, and a script to be executed after installation.  Is this possible? What is the most accurate way to retrieve a user's correct IP address in PHP? Shape of a string/chain/cable/rope/wire? The height of a string in a gravitational field in 2-dimensions is bounded by $h(x_0)=h(x_l)=0$ (nails in the wall)  and also $\int_0^l ds= l$. ($h(0)=h(l)=0$, if you take $h$ as a function of arc length) .  What shape does it take?  My try so far: minimise potential energy of the whole string,  $$J(x,h, \dot{h})=\int_0^l gh(x) \rho \frac{ds}{l}=\frac{g \rho }{l}\int_0^l h(x) \sqrt{1+\dot{h}^2} dx$$  With the constraint  $$\int_0^l \sqrt{1+\dot{h}^2} dx- l=0$$ If it helps, it's evident that $\dot{h}(\frac{l}{2})=0$.  Generally, this kind of equation is a case of a constrained variational problem, meaning that the integrand in  $$\int_0^l \frac{g \rho }{l}h(x) \sqrt{1+\dot{h}^2} +\lambda(\int_0^l \sqrt{1+\dot{h}^2} dx- l)dx$$  Must satisfy the Euler Lagrange equation. The constraint must also be satisfied.  But, in truth, by this point I am clueless. $\lambda$ is worked through $\nabla J  = \lambda \nabla(\int_0^l \sqrt{1+\dot{h}^2} dx- l)$. I have tried this , but get nonsensical answers.  Is this method the best? If so, in what ways am I going about it wrongly thusfar? Can anyone explain how we choose one sock from each of finitely many pairs without the axiom of choice?  I mean : To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.  The idea is that the two socks in a pair are identical in appearance, and so we must make an arbitrary choice if we wish to choose one of them. For shoes, we can use an explicit algorithm -- e.g., "always choose the left shoe." Why does Russell's statement mention infinitely many pairs? Well, if we only have finitely many pairs of socks, then AC is not needed -- we can choose one member of each pair using the definition of "nonempty," and we can repeat an operation finitely many times using the rules of formal logic (not discussed here). I've got a directory shared on my network, using Samba. People on the netwerk use this Samba share to download files from my share to their computers. I'm interested in what files are downloaded most, and by which users they are downloaded. I'd like to see this in real time, if possible. Is there a program that allows me to do this? If yes, how can I use it?    I'm using Ubuntu 11.10. How can I access shared folders on an Ubuntu host and a Windows XP guest?