Prove that there does not exist a group $G$ with a group of automorphisms isomorphic to $\mathbb{Z}$ or $\mathbb{Z}/n\mathbb{Z}$ with odd $n \gt 2$.  I assumed the opposite, let there exist a group $G$ such that $\mathrm{Aut} (G) \cong \mathbb{Z}$ or $\mathrm{Aut} (G)  \cong  \mathbb{Z} / n \mathbb {Z}$. Since the subgroup of the cyclic group is also cyclic, $\mathrm{Inn}(G)$ is cyclic. $\operatorname{Inn} (G) \cong G / Z (G) $.  So $G$ is Abelian.  If $\mathrm{Aut}(G)$ has an element whose order is not equal to $2$, then we take the automorphism $p (x) = x^{-1}$. This is an automorphism of order $2$. There remains a case where the order of each automorphism is $2$. Help with this case.
I had this "almost bonus" question on the final in Group Theory recently: prove that there is no such group $G$ which would satisfy Aut$(G)\cong \mathbb{Z}_n$, where $n$ is an odd integer. I don't have much certainty if this proof is OK, and one's opinion would be appreciated.  Here's my attempt:     Suppose that there exists such a group $G$ that satisfies the above condition. Since $\mathbb{Z}_n$ is cyclic, Aut$(G)$ is also cyclic and $G$ is abelian, which implies that Inn$(G)\cong \{e\}$. Thus Aut$(G)=$Out$(G)$. (I don't think this fact is important here though). Now, since an automorphism sends a generator to a generator, and since each automorphism is completely determined by such mapping, |Aut($G$)| must be of factorial order. But an integer factorial is always even. However, $|\mathbb{Z}_n|$ is odd, which is a contradiction. Therefore, no such $G$ exists.   I'm afraid that my proof is not very nice, but at least a genuine attempt was made.
Is this sentence grammatically correct?  Anyone who loves the English language should have a copy of this book in their bookcase.  or should it be:  Anyone who loves the English language should have a copy of this book in his or her bookcase. Are real guns used in movies? How to get rid of OTA software update notification? I upgraded my Nexus 4 to Lollipop a few days ago but I don't like it. I'll leave out my reasons because they're not relevant here. I wiped my phone and installed a clean stock 4.2.2 version instead. I'm happy with this.  Not long after, Android automatically downloaded the 4.3 upgrade and notifies me with the choices "Later" and "Install". I don't want to do this upgrade.  How do I:   get rid of the notification, and  remove the downloaded update (to free up precious storage space).     I have already tried choosing "Later" but the same notification immediately reappears.    Based on Google search results, I also tried going to Settings &gt; Apps &gt; All &gt; Google Services Framework and unticked "show notifications" but it still appears. Force stop or disable also doesn't seem to help, so I've re-enabled the service including notifications. I didn't see any other apps in this list that sounded relevant - did I just not see it? What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases? Recently I have learned how to draw simple stuff with TikZ. I really like it. Now I wonder if it is possible to draw with TikZ on a image. Instead of pasting a image in Illustrator and adding some vectors to point something in the picture I would like to do it using TikZ. Is it possible?   For example, on a picture of a computer (PNG,etc) I would like to draw some arrows and labels saying "Keyboard", "Monitor", etc. How to initialize all members of an array to the same value? I have a large array in C (not C++ if that makes a difference). I want to initialize all members of the same value.  I could swear I once knew a simple way to do this. I could use memset() in my case, but isn't there a way to do this that is built right into the C syntax? I am creating a type of adventure map. Once you enter the map, you step on some pressure plates that are supposed to seamlessly close the wall behind you. For some reason this command does not work when in a command block or just in text. Here is the command:  /fill -17 4 -109 -18 5 109 minecraft:light_gray_concrete   The output it gives me is "That position is not loaded". I am on a singleplayer superflat world in 1.14.3. Any help would be very appreciated! Scrolling has stopped working in chat My "Meta" rep on Photo.SE differs from my main rep I've been working on some animations recently, and I've noticed that I'm having poor luck with certain things, namely the fact that I don't really use Blender's graph editor very well.  One of the major issues that I have is pretty widespread among 3D graphics programs; when I create an animated object, it defaults to using a curved graph instead of a linear graph, which means that the most movement is at the middle of the animation, and stuff just practically stops near the beginning or the end. While this helps with organic camera movements or such, it's also not what I usually want. Can I set animation curves to default to linear? Auto-inserting Stack Overflow affiliate into all Amazon links We have officially discontinued the Amazon book affiliate remnant ads, since despite our best efforts they don't perform well for our audience and cannot be made to perform well.  We've often wondered,   I'm thinking this isn't a problem; it has gotten lots of support historically. So we're going to go ahead and do it, so the company Amazon account is still getting some use, and we benefit a little.   To be clear: from this point on, all Amazon links posted on all sites will automatically be rewritten to add our affiliate link. We have a clever affiliate redirect that geolocates the IP address and redirects to the appropriate regional Amazon store (amazon.uk, amazon.ca, et cetera).  If you have any feedback on this, or suggestions, as they say &hellip; speak now or forever hold your peace! I want it to know how to convert string field values to integer values in QGIS Dufour... there are numbers and strings in the same field, when I use toreal(string) it stops and it won´t create the field. How to to resize an ext4 partition from the command line? What's the easiest way to resize an ext4 partition (or any type partition depending on the method) from the command line (potentially with the fewest commands, but also the easiest to understand)?  Using a tool like Gparted is obviously easy in a GUI, but what about in the command line? I guess text-based GUIs can count for the answer too since it's technically still in the command line. It just needs to be easy.  By partition I mean a simple partition on a single disk of a personal computer (e.g. on a laptop). For example, I want to resize /dev/sda4. There's no RAIDs, there's not more than one disk drive, there's not anything complicated here. Just a simple partition on a single disk (/dev/sdaX on /dev/sda). Could someone explain to me what the collision of two photons would look like? Will they behave like,   Electromagnetic waves: they will interfere with each other and keep their wave nature Particles: they will bounce like classical balls   I assume that energy of that system is too small to make creation of pairs possible. I bet the automatic response to my question would be "Bell's theorem" and of course I am not disputing Bell's proof. I am however uncertain of one of his assumptions.  The so called "no conspiracy" assumption states that we somehow posses this magical thing called free will which makes us able to break free from the causal chain of events that make us measure the $y$-axis instead of the $x$-axis and so on in experiments. Obviously this makes little sense once reviewed under the light of logic. What this naturally implies is that, if a theory is fully deterministic, it is instantly super-deterministic.  Simon Kochen and John Conway published a theorem in 2009, , which simply-put states: either everything is deterministic (super-deterministic) or every particle has free will.  Among the few who accept this to the fullest extent is Gerard 't Hooft. He has proposed that what we call sub-atomic particles are really just templates and that there is a realm beneath the quantum where the true fabric of reality is "hidden". By accepting determinism he escapes Bell's theorem (due to not having to accept the "no conspiracy" assumption) and can construct a local, deterministic and realist hypothesis of reality. In Gerard 't Hooft's model there is cellular automata somewhere near the Planck scale which gives rise to the "emergent quantum mechanics".  This means that there exist no superpositions in objective reality, the cat is always either dead or alive, no collapse of wave functions, no branching universes, non-locality or retro-causality.  The big question for me is basically: Why do people have such a hard time accepting determinism? Some will object and say "oh but it's SUPER-determinism", but that makes no sense whatsoever, either everything is determined (determinism) or only somethings are (quasi-determinism). By accepting it we escape Bell's theorem. Bell himself was well aware of this and mentioned it a few times in interviews in the 80's.  I know some are concerned that if we accept complete determinism we can no longer do science because it's all a HUGE CONSPIRACY. Gerard 't Hooft has answered critics who bring up this here: , section 6.  Additionally, another author who has taken a deeper look into the worries of "conspiracies" in QM interpretations is Peter J. Lewis in his paper  (also ).  So to all of you who dismiss local hidden variables I ask: Why? Why are local variables not initialized in Java? Was there any reason why the designers of Java felt that local variables should not be given a default value? Seriously, if instance variables can be given a default value, then why can't we do the same for local variables?  And it also leads to problems as explained in :     Well this rule is most frustrating when trying to close a resource in a finally block. If I instantiate the resource inside a try, but try to close it within the finally, I get this error. If I move the instantiation outside the try, I get another error stating that a it must be within a try.      Very frustrating. TAR 6 GB worth of files into separate 1.9 GB TAR files I have 28,000 files of size 6.7 GB in the directory aaa/input. I need create TAR files in the directory aaa/output, such that each TAR file cannot exceed 1.9 GB, so I have to tar the 28,000 files into 4 TAR files (6.7/1.9) as Log_(current date)_01.tar Log_(current date)_02.tar ... etc.  such that each tarball is less than 1.9 GB. Is there a way to do this using a Unix script? When voting to migrate an off-topic question, why is the list of sites limited? Can we make a Vpn server run on Android? Can we make a Vpn server run on Android. Can applications do this? If yes, which protocols are supported and is this a good idea to use such apps? If not, what are some issues that we may face? Autokey (autokey-qt) freezes and won't insert text