I have an exam in two hours and this is one of the exercises in my textbook that I have problem with, I tried solving it using the main theorems but i couldn’t figure it out. The question goes: If G is a nonabelian group with order of $p^3$ which p is a prime number then prove $G’=Z(G)$ A hint or an answer would be appreciated .
Question  If $|G|=p^3$ and $G$ is not abelian show that $G'=Z(G)$  Attempt   Since $|G|=p^3$ then $G$ is solvable and let $$1\leq G^{(n-1)}\leq...\leq G^{(n)}=G,(1)$$  be its derived series.We know that $G^{(n-1)}$ is an abelian subgroup of $G$ so $|G^{(n-1)}|=p$ or $p^2$ (since $G$ is not abelian).Hence we have that $\bigg|\dfrac{G}{G^{(n-1)}}\bigg|=p$ or $p^2$ so $\dfrac{G}{G^{(n-1)}}$ is abelian $\Rightarrow G'\leq G^{(n-1)}$ so $(1)$ is $1\leq G' \leq G,(2)$.   Furthermore, $Z(G)\not=1$ (since $G$ is a $p-$group) and $\dfrac{G}{Z(G)}$ is abelian $\Rightarrow G'\leq Z(G)$  Is this correct so far? How could I continue?
I'm trying to model an , but it's not going so well when it comes to the magazine of the weapon.   Next to the magazine is a mesh. I want to cut holes in the mag with that mesh. But the magazine just goes invisible or connects with the mesh when trying to execute the Boolean Modifier. I have tested to invert the normals and some other stuff, but it isn't working. Is the 64-Bit version of Ubuntu only compatible with AMD CPUs? I was told that computers with more than 2 gig's memory need a 64 bit operating system to utilize all RAM.  Is the 64bit Ubuntu download really JUST for AMD processors? I am asking because the disk image I downloaded says AMD64.  So will my new Intel 2.3Ghz Core i3 Dual Core processor work with 64 bit Ubuntu?  It runs the 64bit version of Windows without any qualm. Transferring Money from India Proof that two basis of a vector space have the same cardinality in the infinite-dimensional case I am having a difficulty setting up the proof of the fact that two basis of a vector space have the same cardinality for the infinite-dimentional case. In particular, let $V$ be a vector space over a field $K$ and let $\left\{v_i\right\}_{i \in I}$ be a basis where $I$ is infinite countable. Let $\left\{u_j\right\}_{j \in J}$ be another basis. Then $J$ must be infinite countable as well. Any ideas on how to approach the proof? can CakePHP automatically create tables from models? Short circuit evaluation and side effects How do I change file permissions on a FAT32 drive? How to select even or odd elements based on class name if you create html layout like so  &lt;ul&gt;      &lt;li class='a'&gt;&lt;/li&gt;      &lt;li class='b'&gt;&lt;/li&gt;      &lt;li class='a'&gt;&lt;/li&gt;      &lt;li class='b'&gt;&lt;/li&gt;      &lt;li class='a'&gt;&lt;/li&gt;      &lt;li class='b'&gt;&lt;/li&gt;      &lt;li class='a'&gt;&lt;/li&gt;      &lt;li class='b'&gt;&lt;/li&gt; &lt;/ul&gt;   and try to select odd elements with 'a' class like so $$('.a:odd') you will get empty array, and if you do $$('.a:even') you will get all four li elements with 'a' class.. It really strange.. But im new to mootools, maybe im doing something wrong..    So my question is how to select first and third li elements with a class. I know that i can do it with function like this  $$('.a').filter(function(item, index) { return index%2; }  but its too complicated for such small task as selecting odd or even elements.. Found this when subscribing to jQuery tag. "unsubscribe | rss" is not shown correctly. Proving that $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ if $n$ is greater than 2 Does it make a difference to CP if you feed candies before or after evolving? When activating the evolution for a caught Pokemon, it seems to amplify the Combat Power (CP). Thus it seems most beneficial to evolve your Pokemon with the highest CP, if you have multiple of the same kind.   But what I am not sure about is whether it makes an efficiency difference if you spend candies and dust to boost CP before evolving, to gain more from the amplification? Or is it more efficient to just evolve without spending, and then using the leftover candies afterwards, since they now give a bigger CP boost per candy?   I only tested with a Rattata so far, which gained +8 CP per candy before evolving, and +15 CP afterwards. I collected no numbers on the amplification efficiency though, and now ran out of materials for further testing.  Or maybe it is a balanced calculation and makes no difference at all? I would love to have some insight on that.    Some data I collected thus far: (GPC = CP gain per candy)  79 CP Weedle (GPC 6) evolved to: 84 CP Kakuna (GPC 6), evolved at 90 CP to: 297 CP Beedrill    Note: The max-CP "half circle" in the profile stayed filled roughly at the same relative position throughout the evolutions. How do I get an RSS feed of my YouTube subscriptions? The default "Subscriptions" feed on the YouTube homepage shows all of the activity of the people that I am subscribed to on YouTube or have in circles on Google+. However, I'm most interested in new videos that the people I subscribe to (either on YouTube or by circling on Google+) create and post (neglecting comments or sharing other people's videos - what you see when you check the "show uploads only" checkbox on the YouTube homepage). However, being able to bring anything off of the YouTube homepage and into Google Reader as an RSS feed would be good.  Is there a way to do this? Let $R$ be a ring such that $x^4=x$ for every $x\in R$. Is this ring commutative? How to include python script inside a bash script Tracking email bounces, opens, clicks A big thank you, Tim Post As you may have seen, this was ’s last week at Stack Overflow. To say that he has been a staple of this company for a very long time is an understatement. Tim has served , evangelist, thought leader, anthropologist, psychologist, and mediator for this site for almost as long as it’s been around. For me personally, Tim has been my right hand in this role. He has educated me on the history of our community, been the person I’ve gone to to ask “why is this thing this way?“, bounced ideas off of, collaborated with, and relied on as a steady voice and a thoughtful teammate. We’re all proud to send him off and support him in any way we can in his next role. Tim has agreed to meet with me once a month to talk these things through moving forward and I’m excited to continue getting his feedback on our initiatives and goals. Thank you, Tim, for being such a stalwart here for so long. We’ll miss you, and look forward to sneaking you into our “Among Us” games for as long as we can. We're also excited to see you request to rejoin the ranks of Stack Overflow moderators and continue to be a pillar in the community. Has any Dark Force user/ally ever said "May the Force be with you"? Set "this" variable easily? I have a pretty good understanding of Javascript, except that I can't figure out a nice way to set the "this" variable.  Consider:  var myFunction = function(){     alert(this.foo_variable); }  var someObj = document.body; //using body as example object someObj.foo_variable = "hi"; //set foo_variable so it alerts  var old_fn = someObj.fn;   //store old value someObj.fn = myFunction;   //bind to someObj so "this" keyword works someObj.fn();               someObj.fn = old_fn;       //restore old value   Is there a way to do this without the last 4 lines?  It's rather annoying...  I've tried   binding an anonymous function, which I thought was beautiful and clever, but to no avail:  var myFunction = function(){     alert(this.foo_variable); }  var someObj = document.body;        //using body as example object someObj.foo_variable = "hi";        //set foo_variable so it alerts someObj.(function(){ fn(); })();    //fail.   Obviously, passing the variable into myFunction is an option... but that's not the point of this question.  Thanks. Is it true that for algebraic sets $V,W$ we have $I(V \times W ) =I(V) + I(W)$? I can’t for the life of me figure out where to use a and where to use the — and where there is no article at all. Is there a simple rule of thumb to memorize?   The standard rule you always hear:     “If a person knows which item you are talking about then use "the"   . . . doesn’t clear things up for me, as I have no idea whether or not they know.