Prove there is no homomorphism from $Z_{16} \oplus Z_2 $ onto $Z_4 \oplus Z_4$   I have no idea at all how to attempt such question. I have read some solutions in which we find the cardinality of $kerm\phi$ for some $\phi mapping $ that is onto using the first theorem of homomorphism.  After that I don't understand anything  Is there a general approach to these questions ?
For some reasons, homomorphism is a very hard area for me to make improvements. I've been hitting the brick wall for almost 2 hours.   Prove that no homomorphism exists from $Z_{16}\oplus Z_2$ onto $Z_4\oplus Z_4$.  Assume a homomorphism $\phi$ exists from $Z_{16}\oplus Z_2$ onto $Z_4\oplus Z_4$. Then $\phi$ is an isomorphism.  Note that the $\left | \ker \phi \right |=2$ The $\ker \phi$ is also a normal subgroup for $Z_{16}\oplus Z_{2}$. We want possible normal subgroup of order 2. I.e., 2 elements in each normal subgroups.  By Lagrange's theorem, the order of each element in a group divides the order of a group so the possible order of the elements are 1 or 2. If the elements has order 1, then the Ker \phi cannot have 2 elements.  Hence, $\ker\phi$ = $\left \{ (0,0),(8,0) \right \}$ ,$\left \{ (0,0),(0,8) \right \}$,$\left \{ (0,0),(8,1) \right \}$, possibly.  By the First isomorphism theorem:  $\phi: Z_{16}\oplus Z_2 \rightarrow Z_2\oplus Z_2$  $\Psi: G/\ker \phi \rightarrow \phi\left ( Z_16\oplus Z_2 \right )=Z_2\oplus Z_2$  $g\ker \phi  \mapsto \phi(g)=\Psi(g\ker \phi)$  Any help is appreciated.  Edit: I know that $(2,0)$ has order $8$ in $Z_{16}\oplus Z_{2}$ but any elements in $Z_{4}\oplus Z_{4}$ does not have order $8$ which would have solved the question at the outset. But I would like a different route.
overset text over equal sign in in an Align environment Look at the following example  \begin{align}      &amp; x \\      \overset{\text{some long explanation}}= &amp; foo \\      &amp; = bar \end{align}   what is the best way to get the equal signs aligned? Now I have seen different solution such as adding \overset{\hphantom{\ţext{some long explanation}}}= to the beginning of every line, but I don't really think that is the best solution. Any better ideas?    UPDATE  I guess what i am actually after is   aligning the first column of the align environment right aligning the equal sign under the overset text right How do I load an mp4 video with multiple audio tracks into blender? During a sunny day the walls of my house warm up (no surprise). My question: how much of this warming up (if any) comes from visible light? I associate infrared with thermal energy. If my house was floating in space (to prevent any thermal exchange with its surroundings) and I installed a giant infrared (and UV) filter between it and the sun, would it still warm up (compared to its rest temperature in full darkness)? Mirror Modifier making ugly smoothing on mirrored faces The model is vivisected right down the middle along the z axis. The right leg is rendering funny and I assume if I can get that fixed everything else will fall into place. I tried flipping normals, changing shading types - can't think of anything else to do. Previously I thought this is a universal theorem, for one can prove it in the one dimensional case using variational principal.   However, today I'm doing a homework considering a potential like this:$$V(r)=-V_0\quad(r&lt;a)$$$$ V(r)=0\quad(r&gt;a)$$ and found that there is no bound state when $V_0a^2&lt;\pi^2\hbar^2/8m$.  So what's the condition that we have at least one bound state for 3D and 2D? How to uninstall Intel Linux Graphics Installer and revert to default drivers I just recently completed a neutral run where I experimented seeing how the story would go on if I only killed certain characters (it was my second run, my first run was True Pacifist) and I ended up choosing to kill Flowey instead of letting him live. I planned to start a Genocide run after resetting the game, but Flowey was gone when i did. Flowey is basically the base for each story in undertale (pacifist and genocide) so I know I can't progress any further without him being here. I really just want to start anew and, without Flowey, I dont think I can access True Resets anymore. It may be possible if I did a genocide route, but then I'd have to fight Sans, and that would take FOREVER.  Basically, I am wondering if I can erase all my progress somehow. I heard that True Resets don't erase everything entirely, so is there another way? Please respond, Undertale is the only game I have that's any fun and I don't want to have to buy another game. As a new user without much rep, what should I do if someone else has already asked a question I have, but it has no answers? Suppose you had a question, and you really want a good answer that would solve your problem, but noticed it was already asked by someone else. However, no one has posted any answers. The question is technically considered "unanswered", but a lot of users who would otherwise bother to give a correct answer probably skip the question completely for some reason.  What should I do in this case?  One idea is to ask a new, completely identical question, in the hope users will notice the new question and post an answer there. But would that go against SE policy?     Note: This is adapted from , but that doesn't cover the case where the original question is unanswered. Also, it is not a duplicate of , as that question asks about getting attention for one's own questions, while this one and the one linked above ask about getting an answer if someone else has already posted the same thing. How can I make a peer-to-peer multiplayer game? How can I make a p2p multiplayer game? I would like to have a server-less multiplayer game. But then, how all the clients know each other?  Why the p2p-protocol is so famous in file transfer but not in multiplayer games? What do you call someone who pretends to be your friend but is actually your enemy?  A friend suggested spy for me, but that does not nearly describe the word I need for an English project. The character is very good at manipulating how people see him. He's a good liar. And pretends to be your friend but actually is your enemy. I was wondering if there is a simpler way to find the last to digits of a power such as $3^{45}$. I reduced it modulo 100 to get the answer, which is 43. But I was curious if there was a simpler, or more eloquent way to get this result.  Here's my method:  Using the fact that $3^4=81\equiv -19 \pmod{100}$ and $19^5=2,476,099\equiv -1 \pmod{100}$  $$ 3^{45}\equiv 3(3^4)^{11}\equiv 3(-19)^{11}\equiv 3(-19)(19^5)^2\equiv 3(-19)(-1)^2 \equiv -57 \equiv 43 \pmod {100} $$  I just felt like this was sort of a messy way to find out that the last two digits are 43. How was this answer posted after this question was closed? Question:   Answer:   The question was closed at 11:57:57 (according to the hover-over-"x-mins-ago" bar) and the answer was posted at 12:16:35. It's a fairly lengthy answer, so it's possible that it was started to be written before it was closed but I swear I've started to answer a question then I got a banner along the top saying "this question has been closed" and the answer was no longer post-able. JQ (GET ID based on NAME) I get a long list of such arrays  {     "id": "byu6g6c4cjys5mdkg5znh8ju8c",     "create_at": 1511875272294,     "update_at": 1511875272294,     "delete_at": 0,     "display_name": "BMA",     "name": "BMA",     "description": "",     "email": "aleksandar.ivanov@random.com",     "type": "O",     "company_name": "",     "allowed_domains": "",     "invite_id": "gdgz1tbxuinntx1fbr1ax7kehy",     "allow_open_invite": false,     "scheme_id": null   }   I want to get by JQ only the ID where the name is BMA. At the moment I parse " jq -r '.[]["name"]" and I can filter the output from curl by name and I will get "BMA" and also 100 other names, but I need to filter only the ID where name is =BMA. Any ideas? Saving Arduino sensor data to a text file There is no continuous injective function $f:\mathbb{R}^n\longrightarrow\mathbb{R}^m$ I have the following question, more general that   Let $U\subset\mathbb{R}^n$ be a domain. If $f:U\longrightarrow\mathbb{R}^m$  ($m&lt; n$) is a continuous function, how to prove that $f$ is not injective.  Any hint would be appreciated. What is divergence? Is it possible to set the default size of the Empty Object? The problem is since I switched my blender scale to metric, the Empty's default size is kinda large for most times I need it. Do you ever need to specify 'javascript:' in an onclick? Why is the speed of light defined as 299792458 m/s? Why is the speed of light defined as $299792458$ $m/s$? Why did they choose that number and no other number?  Or phrased differently:  Why is a metre $1/299792458$ of the distance light travels in a sec in a vacuum? How can I move an installed game from one Steam library folder to another? I use (and have it much bigger) drive D: to store all my user data and games, but I have installed the Steam client on drive C:. When I was installing a first game (Warhammer 40000), I've chosen to place my library folder on drive D: and the game went there.  But now as I've got installed the second game (Portal), I can see that it was put on drive C: despite to the choice I've done previously. When I open Steam - Settings - Downloads + Cloud - Steam library folders, I can see both installation places in the list. Can I merge them into one?  I would even agree to use the default drive C: library (as I've replaced it with a symlink already). When I press to remove one library it says it can only remove empty ones.