Property relationg to non-ableian group $G$ If $G$ is a group in which $(a\cdot b)^i=a^i\cdot b^i$ for three consecutive integers $i$ for all $a,b\in G$, show that $G$ is abelian. Show that the conclusion of above problem does not follow if we assume the relation $(a\cdot b)^i=a^i\cdot b^i$ for just two consecutive integers.   I have solved the first spending about 3-4 hours. However, I cannot come up with the example to problem 2.    Can anyone please help with that?
An example of a non-abelian group $G$ where, for all $a,b\in G$, the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$ I am looking for an example of a group $G$ where the equality $(ab)^{n}=a^{n}b^{n}$ holds for two consecutive integers $n$, but $G$ is not an abelian group. I've started do some calculations in the group $D_{4}$ (I gave up!) Do you know where I can find that example? Is it possible find such example without doing a lot of calculations?  Thanks for your help!
How do I use the conditional (ternary) operator? I've always wondered how to write the "A ? B : C" syntax in a C++ compatible language.  I think it works something like: (Pseudo code)  If A &gt; B    C = A Else    C = B   Will any veteran C++ programmer please help me out? Let $X$ be a locally compact metric space which is also $\sigma$-compact. Let $C_{c}(X)$ be the continuous functions on $f$ from $X$ to $\mathbb{R}$ with compact support. Is $C_{c}(X)$ separable?  My work so far:   If $X$ is a compact metric space, then by Urysohn's Lemma and Stone-Weierstrass, the continuous functions $C(X)$ on $X$ are separable and hence the result follows as $C_{c}(X) = C(X)$.  Suppose $X = \mathbb{R}$. Write $\mathbb{R} = \bigcup_{N  = 1}^{\infty}[-N, N]$. Let $f \in C_{c}(\mathbb{R})$. Then $f$ is supported on a compact set $K \subset [-N, N]$ for some $N$. Thus $C_{c}(\mathbb{R}) =\bigcup_{N = 1}^{\infty}C([-N, N])$. Each $C([-N, N])$ has a countable dense subset $\{\psi_{N, n}\}_{n = 1}^{\infty}$ and so $\bigcup_{N, n = 1}^{\infty}\{\psi_{N, n}\}$ is a countable dense subset of $C_{c}(\mathbb{R})$.  In the general case, $X = \bigcup_{i = 1}^{\infty}X_{i}$ where each $X_{i}$ is compact and $X_{1} \subset X_{2} \subset \cdots$. Let $f \in C_{c}(X)$. Then $f$ is supported on a compact set $K = \bigcup_{i = 1}^{\infty}K \cap X_{i}$. Is it still true that $C_{c}(X) = \bigcup_{i = 1}^{\infty}C(X_{i})$? Converting GeoJSON to Shapefile using ogr2ogr? I have a GeoJson file that I want to convert to ESRI Shapefile using ogr2ogr in FWTools.  I tried using   ogr2ogr -F "ESRI Shapefile" filename.shp geojsonfile.json OGRGeoJSON   I got output      ERROR 6: Can't create fields of type StringList on shapefile layers. ERROR 1: Attempt to write non-polygon (POINT)   geometry to  type shapefile. ERROR 1: Terminating translation   prematurely after failed translation of layer OGRGeoJSON   GeoJSON input we are giving is in standard geojson format.  content of geojson are like  {to share I took only a part from features array }  {"id":1223, "properties":{"name":"3","level":2}, "map_version":11,"type":"FeatureCollection","x_id":1234,"obj_type":"LevelGeometry", "features":[  { "my_area":[131.40093253528894,33.58879853190332,6.515867411008615,3.490549810987222,-0.6237271428108215], "id":1984791, "location":{"type":"Point","coordinates":[131.4009325,33.5887985]}, "properties":{"display_name":"xyz"}, "type":"Feature", "geometry":{"type":"Polygon","coordinates":[[[131.4009508,33.5888314],[131.4008895,33.5887946],[131.4009148,33.5887653],[131.4009761,33.5888020],[131.4009508,33.5888314]]]}, "obj_type":"Geometry" }  ] } Inspired by this ....  I am the sole person using my system with 12.04. Every time I issue a sudo command; the system asks for the user password (which is good in its own way).  However I was thinking; without activating the root account; how can I execute the sudo commands which will not ask for user password to authenticate.  NOTE: I want to execute sudo command without authenticating via password; only when they are executed via terminal. I don't want to remove this extra layer of security from other functions such a while using 'Ubuntu software center' or executing a bash script by drag-drop something.sh file to the terminal. Why does an overridden function in the derived class hide other overloads of the base class? How can one change the timestamp of an old commit in Git? The answers to  describe a way to amend previous commit messages that haven't yet been pushed upstream.  The new messages inherit the timestamps of the original commits.  This seems logical, but is there a way to also re-set the times? How does a zener diode and a resistor regulate voltage? Is there a way to have Cycles skip rendering alpha? If I wanted to create an animation, each frame would have several seconds of rendering a completely transparent part, which adds up obviously. Is there a way to eliminate this waste of time? I have been reading Stack Exchange for years - mostly not logged in, and have never posted (this is my first). I'm simply not very talkative, and very good at googling.  Recently a few occasions have come up when I've wanted to comment on a question.  I wanted either to ask for, or provide, clarification which did not warrant a new answer.  How do I gain enough reputation to comment without:   asking questions which have already been answered or that I can figure out on my own, or posting my comment like as an answer (i.e. being  rude)? The observable universe is approximately 13.7 billion years old. But yet it is 80 billion light years across. Isn't this a contradiction? Proving that surjective endomorphisms of Noetherian modules are isomorphisms and a semi-simple and noetherian module is artinian. I am revising for my Rings and Modules exam and am stuck on the following two questions:  $1.$ Let $M$ be a noetherian module and $ \  f : M \rightarrow M \  $  a surjective homomorphism. Show that $f : M \rightarrow M $ is an isomorphism.  $2$. Show that if a semi-simple module is noetherian then it is artinian.  Both these questions seem like they should be fairly straightforward to prove but I cannot seem to solve them. Compiler Errors in atlwin.h Why is observing 100 heads for a fair coin flips surprising? Assume that we have a fair coin. We flip it 100 times. The outcome is all heads. Why is it surprising? Doesn't all outcomes have the same probability? Any particular outcome, including irregular ones, would have the same very small probability. Why is it that observing an irregular outcome is less surprising to us than a regular one? Is it correct to use Bernoulli's theorem to explain that a disk floats in the air? Should the correct explanation be based on Boyle's law? In this video, a device is introduced from 1 minute to 18 seconds. After entering the air, the disc at the bottom of the device will be suspended in the air. The video explains that this is because of the Bernoulli effect, but I think this explanation is incorrect, because the disc gap, the radial volume is increasing. Therefore, the reason why the disk is suspended is that the air volume becomes larger, so the pressure decreases (according to Boyle's law). If the air moves along the radial direction at a constant speed, the volume of the air must be constantly increasing, right? So the pressure is decreasing, right? Citation: refer to author or paper? What is the British English way of referring to a source in-text.  Is it plural because there are two authors, like this:         X and Y (2011) describe ...   Or is it singular because you are referring to a singular source, like this:     X and Y (2011) describes...   So what I am asking is whether you refer to the source or to the authors? When do mobs despawn in Minecraft? I understand monsters only spawn in a 9 x 9 grid of chunks centered on the player, and I assume they stay around while they remain within that distance of you. But do they despawn as soon as you move away from them? Or do they remain? If so what (if anything) causes them to despawn? Are the rules any different for passive mobs?  Will running far away from an area and then coming right back clear it of monsters? How to use a field value in Template.php Redefining "cited on" string (and others) in biblatex Is there a symmetry associated to the conservation of information? Conservation of information seems to be a deep physical principle. For instance, Unitarity is a key concept in Quantum Mechanics and Quantum Field Theory.  We may wonder if there is an underlying symmetry, in some space, which may explain this conservation of information. Is there an easy way to log all commands executed, including command line arguments? I am trying to find how to log a specific instantiation of rrdtool to see whether the path it is receiving is incorrect.  I know I could wrap the executable in a shell script that would log the parameters, but I was wondering if there was a more kernel-specific way to monitor for that, perhaps a filesystem callback that sees when a particular /proc/pid/exe matches a given binary?