I'm working on a rather tough question (well, at least for myself):     Show that no commutative group of composite (or infinite) order is simple.   Any ideas on how I should approach this? All I know is that groups that contain no proper normal subgroups are simple groups. How can I use this to generalize it to any commutative group? Any hints/suggestions are welcome, since that is all I am able to come up with, at the moment at least.
Just wanted some feedback to ensure I did not make any mistakes with this proof. Thanks!  Since $G$ is abelian, every subgroup is normal. Since $G$ is simple, the only subgroups of $G$ are $1$ and $G$, and $|G| &gt; 1$, so for some $x\in G$ we have $x\neq 1$ and $\langle x\rangle\leq G$, so $\langle x\rangle = G$. Suppose $x$ has either infinite order or even order. Then $\langle x^2\rangle \leq G$ but $\langle x^2\rangle \neq \langle x \rangle$, a contradiction. So $x$, and therefore $G$, has odd order, and by Feit-Thompson, $G\cong Z_p$ for some prime $p$.  Edit: Thanks, I see that Feit-Thompson is too much.  Since $G$ is abelian, every subgroup is normal. Since $G$ is simple, the only subgroups of $G$ are $1$ and $G$, and $|G| &gt; 1$, so for some $x\in G$ we have $x\neq 1$ and $\langle x\rangle\leq G$, so $\langle x\rangle = G$. Suppose $x$ has infinite order. Then $\langle x^2\rangle \leq G$ but $\langle x^2\rangle \neq \langle x \rangle$, a contradiction. So $x$, and therefore $G$, has finite order. Suppose $x$ has composite order $n$ so for some $p &gt; 1$ that divides $n$, $\langle x^p \rangle$ is a proper non-trivial subgroup of $G$, so $G$ is not simple. So $G$ is a cyclic group of prime order.
The escape velocity of earth is roughly $11 kms^{-1}$. However, what if a long ladder was built extending out of Earth's atmosphere and considerably more. Then if something was to climb up at much less than the escape velocity, what would happen when it reached the end?  And what if the object that climbed the "ladder" then fired some kind of thruster/rocket and was going fast enough so that it orbited the Earth. It would mean less energy required to get into orbit? How do I create a completely unattended install of Ubuntu? I need a CD or USB stick, that will install Ubuntu on a completely headless machine. Stick CD in and restart, no keyboard or screen should be involved.  There are a few obstacles:   The language selection menu when you first boot the CD. The fact that the CD menu waits. The installer's asking questions during installation. Can Eigenvalues[] and Eigenvectors[] be assumed to return the same ordering? Is "funnest" a word? 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These are all questions to help me get a better feel for C#.     private string Tabs(uint numTabs) {     IEnumerable&lt;string&gt; tabs = Enumerable.Repeat("\t", (int) numTabs);     return (numTabs &gt; 0) ? tabs.Aggregate((sum, next) =&gt; sum + next) : "";  }    private string Tabs(uint numTabs) {     StringBuilder sb = new StringBuilder();     for (uint i = 0; i &lt; numTabs; i++)         sb.Append("\t");      return sb.ToString(); }    private string Tabs(uint numTabs) {     string output = "";     for (uint i = 0; i &lt; numTabs; i++)     {         output += '\t';     }     return output;  } Best way to add KeyEvents and faster Quit I can describe "IMHO" as an acronym for "in my humble opinion."  How can I describe the reverse processes of translating the acronym into the actual phrase?  That is to say, if "IMHO" is the acronym, what do we call "in my humble opinion"? How can I change the umask for one command only? I’m building a battery powered data logger with a microcontroller and several sensors. I’d like to measure the supply current of individual parts with idle currents in the μA range and peak currents in the low mA range. The current is not constant which rules out a simple multimeter.  Usually I’d (temporarily) replace ferrite beads on the PCB with a 10Ω resistor and measure the voltage drop with my DS1104Z oscilloscope. However, the voltage drop of some μV for 1 or 2 μA of current is simply drowning in the noise and peak currents of ~10mA already result in a noticeable supply drop (so increasing the resistance is out of the question).  I was thinking about building a small board with a 5Ω resistor and OPAMP set to something like 100x amplification, but first wanted to know if this is a good idea or if you know of any affordable, ready-made solutions. Obviously the bandwidth doesn’t have to be high, but some accuracy would be desirable.  Thanks in advance.  Michael "Show desktop" keyboard shortcut in Mac OS X I am using a MacBook Pro with Mac OS X 10.5. Is there a similar function to "show desktop" (shortcut is WIN+D) on a Mac?   It should hide all applications and show a clean desktop. Is there a detailed activity log on Mac OS X that lists my recent commands and actions? How can I see what I just did on my computer? For example, an activity log showing that I copied a file from here to there? Ubuntu 19.10 change desktop environment I've installed another desktop environment which I would like to try. I am not sure how to switch to it.  Is there still a button on the log in screen that will switch desktop environments? This same question for previous versions of ubuntu yields screenshots of this button. But I cannot find any button on the log in screen that does this.  So, in ubuntu 19.10, does there exist a button to switch desktop environments? If so, where is it? 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