I have a group of order $p^\alpha$, $p$ is prime. I argued that by the Lagrange's theorem, the subgroups of this group are of prime power order. Just want to clarify if this also implies the existence of these subgroups? Can I say this group has  subgroups of order $p$, $p^2$, ..., $p^{\alpha-1}$? Or do I need to refer to other results/theorems? Thank you.
Prove that if $|G| = p^n$ then $G$ has a subgroup of order $p^m$ for all $0\le m &lt;n.$   Since $G$ is of prime-power order I know $|Z(G)| \ne e$ so there is an $a\in Z(G)$ with order $p$ such that $p \mid |Z(G)|$. Now, the subgroup generated by is normal since it's a subgroup of the center. How can I get this normal subgroup to be less than $n$?
If a sequence $(u_n)$ be such that its every subsequence has a subsequence that converges to $0$, then $\lim u_n= 0$ Does anyone know if there are any video formats (besides the HUGE uncompressed avis) that you can render in that maintain the alpha channel? (i.e. for after FX etc.)   I used to use quicktime, but that's no longer an option.  I know you can render in png. or other image formats, but video is significantly smaller, and I often work on machines with less space &amp; resources. Doesn't Matlab work on ubuntu 16.04? How can I get the active screen dimensions? How to exactly convert an arithmetic rgb value (e.g 0.55,0.25,0.1) to the corresponding 8bit RGB value (e.g. 123,90,40)?  My more specific questions are:  1.) What is the exact conversion formula and macro?  2.) Is there some built in command or macro for converting it? While adding to an Answer to  question, I needed to use the above phrase, and I suddenly realised I was unsure whether to write "is" or "are".   There is more than one way to skin a cat. If there are more than one species of cat, we will flay each species differently.   I don't think that second example above sounds quite right to me (disregarding the meaning - sorry!), but in my original Answer I felt "is" would have been even worse.  Is "are" always wrong here? If so, what exactly is the rationale? If not, is there a case where it's unquestionably preferred – and why is that?    Note that in both the problematic Answer and my example attempting to mimic the same context more briefly, the writer doesn't know how many there are. There may be one, more, or even none.  Getting even more specific, the writer might have an opinion on how many there might be, and wish to convey his leanings by choice of verb form if language allows this to be done succinctly. Constructing an explicit isomorphism between finite extensions of finite fields Suppose $K$ is a finite field, $K = \mathbb F_{p^s}$. If we take an irreducible polynomial $f$ of degree $d$ over $K$, then the splitting field $L$ of $f$ is $K(\alpha)$ where $f$ is the minimal polynomial of $\alpha$. But then $L = \mathbb F_{p^{sd}} $. Since $\mathbb F_{p^{sd}}$ is unique, we see that this is the splitting field of every irreducible polynomial of degree $d$ over $K$.   Take $K = \mathbb F_2$ and let $P(X) = X^3 + X + 1$, $Q(X) = X^3 + X^2 + 1$. Let $L$ be the splitting field of $P$ and $L&#39;$ be the splitting field of $Q$. The above tells us that $L$ and $L&#39;$ are isomorphic. I would like to construct an explicit isomorphism between $L$ and $L&#39;$.   I know that $L \cong \mathbb F_2[X] /(X^3 +X + 1)$ and $L&#39; \cong \mathbb F_2[X] / (X^3 + X^2 + 1)$. Intuitively, I want to find an isomorphism $\phi : \mathbb F_2[X] \to \mathbb F_2[X]$ such that $\phi((X^3 + X + 1)) = (X^3 + X^2 + 1)$. A little playing around gives me $\phi(X) = X+1$. It now feels like I'm falling at the last hurdle: how do I finish the construction of an isomorphism between $L$ and $L&#39;$? I don't think $\phi$ makes sense as a map from $L$ to $L&#39;$, yet it seems the map I want. How Do I Opt Out of Privileges? According to  , gold badge holders can now unilaterally close questions as duplicates. I hold seven gold badges for tags on SO, which means I can, on my own, close questions as duplicates for those tagged questions.  Simply put: DO NOT WANT.  However, I do not see any ability in SO (the only SE site I spend much time in) to opt out of any privileges, let alone this one.  Is there a spot that I'm just not seeing that allows me to turn off this new privilege? What is the relation between the Higgs field and chirality? Wikipedia  that the spontaneous breaking of chiral symmetry "is responsible for the bulk of the mass (over 99%) of the nucleons".  How do the nucleons gain mass from the spontaneous breaking of chiral symmetry? Why don't leptons gains mass from it? What is the role of the Higgs field in this all? Given $f(x) = \frac{1}{2}x^TAx + b^Tx + \alpha $   where A is an nxn symmetric matrix, b is an n-dimensional vector, and alpha a scalar. Show that  $\bigtriangledown _{x}f(x) = Ax + b$   and  $H = \bigtriangledown ^{2}_{x}f(x) = A$   Is this simply a matter of taking a derivative with respect to X, how would you attack this one? launch an exe/process with stdin stdout and stderr? I'm unable to update nodejs, and I can't see why.  If I were to make a guess, I'd say that I have two repositories listed in apt-get, that both contain a nodejs package.  Here's what I've done:  Remove old version:  sudo apt-get remove --purge nodejs npm   Then, going by , I've run:  curl -sL https://deb.nodesource.com/setup_5.x | sudo -E bash -   And then:  sudo apt-get install -y nodejs   I then run:  nodejs -v   Which returns:  v0.10.25   Can anyone explain what's going wrong? And how I can remedy this situation?  Many thanks. Can you stay in Turkey for over 90 days within the 180 day visa period but split over different visits? The Turkish e-visa is valid for a period of 180 days and allows you to stay for a maximum of 90 days. Am I allowed to stay for say 50 days, then another 50 days, over two different visits within the 180 day period?  I'm looking at the e-visa website  and it doesn't seem to be too clear on this. I am using ArcMap 10.2 and I'm looking for a way to add  a table containing certain fields from select elements from a featureclass to my map. Ive tried the add table option from the attribute table, but it doesn't quite give what I want (the table must be editable to look a certain way).   Are there any options out there for 10.2 like arcpy.mapping.tableview or will upgrading to 10.3 so I can get access to the "graphic table element" function be best? How do I clone a generic List in Java? Is there a complete list of these? I've seen lots of Q&amp;A about "What does x icon mean" but it would be really useful if there were a complete list for reference! Why does subsurf modifier cause weird deformation at the end of a cylinder? My problem is simple. I think that the strange deformations on the top are there because there are so many vertices in one face. How do I fix this while maintaining nice topology?  Although I triangulated the top polygon, the problem still persists. How to convert decimal to hexadecimal in JavaScript I'm new to field extensions and I can neither see why or how to prove this statement.  If I'm not mistaken, to construct a splitting field, you construct rupture fields one after the other, using that the $8$ element field is the splitting field of $X^8-X$ over $\mathbb F_2$, since $2$ is the only prime divisor of 8.  Over $\mathbb F_2$ $X^8-X$ has two roots : $0$ and $1$. Therefore $X^8-X=X(X-1)(1+X+...+X^6)$. The last factor, call it $P$, has no roots. So let $\alpha$ be a root in some extension $\mathbb F_2(\alpha)$ of $\mathbb F_2$.  We now have $P=(X-\alpha)(X^5+\alpha X^4+...+\alpha^4X+\alpha^5)$.  I suspect $\alpha$ might not be the way to go and if so, my second plan would be to reduce $P$. I don't know how to do this but assuming it to be done, hence having $P=P_1...P_n$, I could start using $P_1$'s rupture field, and then my best guess is that if $\mathbb F_2[X]/(P_1)$ is not sufficient, I re-reduce $P$ over this new field and consider the rupture field of one of the factors, and so on.  Now assuming I wasn't given the statement itself, is it elementary to construct the $8$ element field using $\mathbb F_2$ ? Consider this example from the Rogue class traits:     Trained Skills: Stealth and Thievery. From the class skills list below, choose four more trained skills at 1st level.   &nbsp;&nbsp;&nbsp;&nbsp;Class Skills: Acrobatics (Dex), Athletics (Str), Bluff (Cha), Dungeoneering (Wis), Insight (Wis), Intimidate (Cha), Perception (Wis), Stealth (Dex), Streetwise (Cha), Thievery (Dex)   The Rogue is trained in Stealth and Thievery. However, they're also included in the Class Skills section, and it invites you to: 'choose four more trained skills' from that list. This seems to be similar for all the classes.  So can you train Stealth and Thievery as a choice as well have it initially trained or the equivalent for other classes?