Are there groups $G$ and $H$ such that $G$ and $H$ are not isomorphic but $G \times \mathbb Z$ and $H \times \mathbb Z$ are?
I already know that if groups $G,H,K$ are finitely generated abelian groups, following is true.  If $G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$. I prove this by uniquness of factorization of finitely generated abelian groups.  My questions are   If group $G$ is finite,$G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$? Can you give me a easiest proof and it can be proved by projection function on external direct product? If $G,H,K$ are groups.  If $G\times K$ is isomorphic to $H\times K$, then $G$ is isomorphic to $H$. This statement is false. What is a counterexample?   *$\times$ is external direct product
What causes Nova's Triple Tap to go on a short cooldown? This isn't about the level 20 ultimate talent.  There is a brief "arming phase" where Nova locks onto her target before shooting three bullets.  When something happens to her target during that phase (allied Devouring Maw, Divine Shield, etc.), sometimes Triple Tap is placed on the full 100 second cooldown, other times it is placed on a much shorter 10 second cooldown.  What triggers this 10 second cooldown?   Is this a timing issue?  I believe if she is interrupted after the shots start going out, Triple Tap is always put on the full cooldown.  Maybe I'm just seeing the ability get blocked right as the first shot goes out? Is this ability-specific?  Does stunning Nova have a different effect than a Devouring Maw or Void Prison on her target?  And is that different from a Divine Shield/Sanctification on the target? Is this determined by whether the line to the target was interrupted by an ally or an enemy hero?  For example, is there a difference between an allied Void Prison and an enemy Void Prison on the target? How do you Set Custom Time in Ubuntu? Prove that the set of palindromes are not regular languages using the pumping lemma. Firstly I pick a language $xyz$ where $x = \epsilon$, $y = (abb)^{k}$, $z = (bba)^{k}$ where $|y| \ge$ the number of states in the automaton representing my language. Then $xyz = (abb)^k(bba)^k$ is a palindrome.  Now I split $y$ into $uvw$, where $v \ne \epsilon$ contains a state visited more  than once. If the language is regular then $xuv^iwz$ is within the language $\forall i \ge 0$.  I choose $i = 2$, then $uv^2w = (abb)^n$ where $n &gt; k$.  Then $xuv^2wz = (abb)^n(bba)^k$ where $n &gt; k$, which is not a palindrome.   Firstly, is my reasoning for this particular proof correct?  I'm unsure about how to properly apply the pumping lemma to this problem. And secondly the question asks me to state a proof for palindromes in general, whereas I only attempted to prove it for a single case. Is there an example I can choose which proves the conjecture for all palindromes? I assume not, seeing as I can't think of how that could be represented using regular expressions. My Custom Skin won't Change I shifted to Ubuntu 10.10 now I am facing problem to have C/C++ compiler. What are the possible ways so that I can use C/C++ compiler using all the libraries (like graphic, math, conio, stdlib, etc) as in Microsoft Windows creating executable file? Notice these sentences:   Shops are open late in summer.   Summer is a traditionally viewed as a slow season for games.   At the summer solstice, the days are longest and the nights are shortest.   it's time to start thinking about what 2016's Song of the Summer will be.    In English class, they told us that "the" is used for the things that are known.  One example for this was a phrase like "The man who killed his wife".  So when we are speaking of that man, we know what he did. Somehow we know something more than nothing about him. He's not just every man in the world.     So Now I'm confused about why "the" is not used in the first and second sentences. Can anyone please explain it for me? Why does water pouring from a glass sometimes travel down the side of the glass? ArcGIS Serious Application Error I'm a complete novice to GIS and just began an Introductory course. I am having an issue when I try to access the Symbology tab on a layer. Specifically, when I click on the 'Categories' field and 'Unique Values' subfield, ArcMap crashes and comes up with a report window to send to ESRI. The message is      ArcGIS for Desktop has encountered a serious application error and is   unable to continue.   What could be the problem? Does pointwise convergence of continuous functions on a compact set to a continuous limit imply uniform convergence on that set? The problem is in Marsden - Elementary Classical Analysis 2nd ed. Ch.5.  My approach was   Condition : $f_k, f(\in\mathcal C) : A\to N$, $f_k \to f \ \text{(pointwise)}$ and $A$ is compact. Since $A$ is compact, the range of $f_k, f$ are all compact, so bounded. Since $f_k, f\in\mathcal C_b$, if $\|f_k-f\|\to 0$ in $\mathcal C_b$ then $f_k\to f \ \text {(uniformly)}$.   when the norm of $\mathcal C_b$ is defined by $\|f\|=\sup\{\|f(x)\||x\in A\}$.  I got stucked in 3, showing $\|f_k-f\|\to 0$. I don't even know the proposition is true. Can anyone give me some hint to solve this or a counterexample for this? I am going to install Ubuntu to dual boot with Windows 8 soon, but one of my friends warned me that partitioning my hard drive will slow my computer down. I run on a 500GB HDD and 8GB of ram if that helps at all. Is that true if I do it? Cannon Locations in New Super Mario Bros. Wii? In Super Mario Bros. Wii, there are cannons on the overworld map that you can use to skip worlds. However, the cannons are initially inaccessible, and you need to complete certain levels in a specific manner to get to them. What levels do I need to complete in order to access these cannons? Lightning:input type="date": how to change the date fomat - Summer'18 Let $X_1,...,X_n$ be mutually independent RVs. Suppose $X_i \perp \mathcal F $ for $1\ \leq \forall i \leq n$.    How can I show that:  $$\sigma (X_1,...,X_n) \perp \mathcal F$$ ?  What I have tried:  $\sigma (X_1...X_n) = \sigma(\cup_{i=1} ^n \sigma(X_i))$ holds. I think I need some Dynkin's lemma-ish argument but can't see how. How to convert Bedrock texture pack to Java Edition Minecraft I'm trying to use a texture pack designed for the Bedrock Edition of Minecraft, on Java Edition. I looked at the texture names for a Java Edition texture pack and compared those to the names of the Bedrock Edition texture names, and they aren't the same. A lot of the texture names are flipped, such as:   Jungle Log (Java = jungle_log.png) (Bedrock = log_jungle.png)   Most of the texture names are flipped, like the Jungle Log (and all of the other log types) and there are some .json files that are different and/or not there.  Trying to open the Bedrock texture pack would result in some animated blocks not being animated, other blocks play the wrong sound (or no sound) when broken, if OptiFine is in the mix, the skybox could be missing, which I am pretty sure would result in an always black sky, along with no sun and moon textures.  Any help in finding a program or site that can do this would be helpful, because I tried doing it manually, but almost every texture is different.  Any help is appreciated! It happens sometimes to receive or see a very good answer that arrives very late and that would deserve more upvotes. What can be done to give a boost to them? I would happily spend some of my reputation points to be able to give them such boost. This is not merely a question of reputation, but the idea is to let the answer &quot;bubble up&quot; faster so that it can be seen easily by other people to look into question. I hope I explained my question so that is understandable. Most of you got the point and there are some interesting proposals like the one that proposes to put randomly an answer at top. But I do have not enough experience to say if there is one that could solve the issue or not. I would be curious to know what Jeff thinks about it, if it feels there is something that can be improved or not. Where do I find the maximum node? A lot of people recommend to use the maximum node for glass setup, but I can't find that node. Nothing comes up when I search for it.  Where do I find it? What is the person called who always thinks negative? E.g.        Mr. X always thinks negative.   Mr. X has negative attitude.      What we can use either? Is there any particular word for this? Prove $GL_n(\Bbb{R})/SL_n(\Bbb{R}) \cong \Bbb{R}^\times$ $$GL_n(\Bbb{R})/SL_n(\Bbb{R}) \cong \Bbb{R}^\times$$  This is trivial to prove with the first isomorphism theorem - by using the determinant as a endomorphism, then as $SL_n(\Bbb{R})$ is $1$ under the determinant, it is the kernel, and by FIT, the above is proved.  I was wondering how to prove this without the isomorphism theorem ? Outline around text on CMYK black background in Photoshop I'm trying to make a document with one color black  (c:0,m:0,y:0,k:100) background with red text on it. However in CMYK the text has this weird grey white-ish edge. In RGB it doesn't show but in CMYK it shows. I'm wondering will this show when printing or do I need to use different methods? ie. using rich black etc.  Here's the RGB and CMYK comparison piping in real time This is more a conceptual thing than a current problem.  Would it be possible to do this:  tail -f &lt;file&gt; | grep "string" | sed 's/stuff//g' &gt;&gt; output.txt  The problem I'm thinking is that tail -f never terminates so it won't progress in the command. I'm trying to read and output a file in real-time. If this doesn't work how would I do it?