I need a proof of the following fact: if a group $G=M \times N$ is the direct product of subgroups $M$ and $N$ such that $\lvert M \rvert$ and $\lvert N \rvert$ are relatively prime, then $\mathrm{Aut}(G) = \mathrm{Aut}(M) \times \mathrm{Aut}(N)$. This is the way the result is recalled on some lecture notes I am reading, I think $M$ and $N$ are implicitly supposed finite (maybe $G$ also?) otherwise this does not make sense. This should be rather standard (it is used but not proven ), but I've taken a quick look on Rotman, Roman and Robinson but none of them seems to prove this. Do you have a reference for the proof/can you share here a proof you know? Thanks
I was working on a problem in group theory, which asks about the automorphism group of a direct product of groups.  Okay, so I know that if $G,H$ are two groups whose orders are relatively prime, then $\operatorname{Aut}(G\times H)\cong\operatorname{Aut}(G)\times \operatorname{Aut}(H)$.  I also know that if $G,H$ are abelian and simple and isomorphic, then $\operatorname{Aut}(G\times H)\cong \operatorname{GL}_2(\mathbf{Z}_p)$ where $p$ is the order of $G$.  What I don't know is the following: what if $G,H$ are simple, yet not abelian? Let us for now focus on a more restricted case where $G\cong H$, in particular, $G=H$. What can we say about the group $\operatorname{Aut}(G\times H)$? (I heard something about this group being related to what is called "wreath product", which I do not know...)  Thanks bunch in advance!
This is what we are taught: $$5^{-2} = \left({\frac{1}{5}}\right)^{2}$$  but I don't understand why we take the inverse of the base when we have a negative exponent.  Can anyone explain why? Word for the situation of being unable to pass opposing pedestrian, as you both start to step same direction Is there a word or expression in English, which describes the situation, when you can't pass a  stranger, who is walking towards you on the street, because you both start to step the same direction?  I'm pretty sure I've heard one once. As I remember, it was including "dancing" reference, but in more specific way. What factors determine enemy levels or: How do enemy levels scale to the player character's? When I finished Normal Mode at around Level 30 and went back to the  (still at Normal mode), I found that all enemies there are now very easy to kill, meaning they didn't level up (or hit a level cap).  When I went to  as a Level 36 character in True Vault Hunter Mode, I noticed that the enemies are the same level as me.  What factors determine enemy levels?   Specifically, I would like to know:       Are enemies affected by what level a character accepted a mission?      Or is it what level a character is when they first entered an area?      How does Normal Mode, True Vault Hunter mode, and the Ultimate True Vault Hunter Mode affect enemy levels?     Are there enemies that will scale to the player level, as the player levels up? Any enemy leveling differences specific to the DLC areas?         EDIT:   There is now  that explains how this works generally: "In playthrough 1, outside of the DLC, there's no such thing as scaling. Enemy levels are determined solely by the area they're in and your mission." Can anyone expound on that, and comment or provide an answer on what are the specific enemy levels on specific areas and/or missions?    states that "enemies do not actually "scale", or at least from what we know of..." and lists specifics for enemy levels on certain areas in the game (sans DLC areas). Can anyone expound on that and/or add specifics for enemy levels in the DLC missions or areas in TVHM (PT 2.0) and/or UTVHM? The  update has just been released, can anyone provide info about enemy levels on that mode, including the DLC areas or missions? Correct the "about" page to reference the new "on hold" policy I frequently need to do the following things in MySQL:   Create a non-root user Set that user's password Grant that user all privileges Make a database with the same name as the new user Allow usage only for users from the localhost.   Previously I did this using PHPmyadmin, but I would prefer doing it directly from BASH. Is there a CLI way to execute these steps? Cannot authenticate without a password Finding the unique rock with its weight I have 12 rocks, all have the same weight except for one rock, but I don‘t know which rock is. How do I know which rock is different from other, and also know that rock heavier or lighter, with maximum 3 comparating steps. My only tool is just a balance that can only compare the weight between the rocks. I am using LaTeX with the article document class. The Bibtex bibliography style that I am using is agsm, part of the Harvard family of bibliography styles.  I have two questions:  (1) If I cite an entry of type @book, then in the reference list I get publisher, address. I would like to re-arrange this order to address: publisher. Which part of agsm.bst should I modify?  (2) I would like to change single quotation marks ' ' to double quotation marks " " for setting off the title of entries of type @article. Which part of agsm.bst should I focus on? Suppose that $U_n=\frac{1}{\sqrt{2n\sigma^2}}\left(\Sigma X_j-\Sigma Y_j\right)$,  where $X_1,X_2,\ldots$ and $Y_i,Y_2, \ldots$ are i.i.d. sequences of random variables with mean $\mu$ and variance $\sigma^2$ and the sequences $\{X_i\}$ and $\{Y_i\}$ are independent.  Then, for all real numbers $x$, show that, as $n\to \infty$,  $$\mathbb{P}(U_n\leq x) \to  \int_{-\infty}^{x} \frac{1}{(\sqrt(2\pi))} \:e^{-\frac 12{u^2}}\mathrm du $$ Approach  I tried to carry out the integration on the right, but this does not integrate nicely. It ends up involving an error function, so I figured this was the wrong was to go.  Should I be looking at convergence in probability to answer this question? Is Chebyshev's inequality relevant here? If so, how and how does that relate to the integral? What information does the trace of a matrix give? My only problem is making them line up three-across and have equal spacing. Apparently, spans can not have width and divs (and spans with display:block) don't appear horizontally next to each other. Suggestions?  &lt;div style='width:30%; text-align:center; float:left; clear:both;'&gt; Is what I have now. On my Toshiba Encore 8 Tablet I noticed that the CPU usage is very high. In Taskmgr / ProcessExplorer, I can see that OneDriveSetup.exe is hogging 1 CPU cores heavily for hours.    How an I solve this hang and finish the OneDrive Update? setSystemUiVisibility(SYSTEM_UI_FLAG_LAYOUT_HIDE_NAVIGATION) does not work Pumpkin Noodles is a thing or are a thing? Which of the following is correct?  How come pumpkin noodles is not a thing?  or  How come pumpkin noodles are not a thing?  I want to post one of the above statements to Twitter with a picture of a bowl of pumpkin noodles i.e. the stuff that comes out of a hollowed out pumpkin. In the  library, they are overloading the () operator.  Is this a convention in C++? For callbacks, etc.?  I have seen this in code of a co-worker (who happens to be a big Boost fan). Of all the Boost goodness out there, this has only led to confusion for me.  Any insight as to the reason for this overload? Initializing K-means centers by the means of random subsamples of the dataset? If I have a certain dataset, how smart would it be to initialize cluster centers using means of random samples of that dataset?  For example, suppose I want 5 clusters. I take 5 random samples of say, size=20% of the original dataset. Could I then take the mean of each of these 5 random samples and use those means as my 5 initial cluster centers? I don't know where I read this but I wanted to know what you guys think about the idea.    UPDATE: Please see this thread  for the general discussion about the various initialization methods. Subfiles failing with custom class The value of a, b and c for this integral Find out the value of $a, b$ and $c$, for which $$ \lim _{x\to 0}\:\:\frac{1}{x^5}\left(\int _0^x\:\left(a+b \cos\left(t\right)+c \cos\left(2t\right)\right)dt\right)=\frac{1}{5} $$  My first thought was to separate the integral into multiple ones because i thought that it is pretty clear that each integral is a common one and a,b,c are constants and can be taken out . After I solved the integral, that's where I had the problem. I couldn't do the limit of the result that I've got in order to find out the value of $a,b$ and $c$ . nmake fatal error U1034: syntax error : separator missing What changes with Kerberos authentication in IPv6 when everyone has a public IPv6 Address? How should Kerberos authentication be set up with IPv6?     What implications are there when client devices each use a public, globally routeable IPv6 IP address?  What changes when setup this way instead of using a NAT as was the norm with IPv4?