I think this proof is a bit sloopy, but the idea works right? If $G/Z(G)$ is cyclic, then $G$ is abelian. Let $G/Z(G)$ be the quotient group with center $Z(G)$. If $G/Z(G)$ is cyclic, then $G$ is abelian.   I was thinking that since $G/Z$ is cyclic, this implies the induced projection isomorphic quotient map $\phi : G/Z \to G$ would give the result. So such a $\phi$ is given by say $$\phi(gZ(G)) = gz$$ for $z \in Z(G).$  I feel like I made an assumption that is incorrect. I looked up the canoical homomorphism and it appears it usually is a map $\pi : G \to G/Z$.
If $G/Z(G)$ is cyclic, then $G$ is abelian Continuing my work through Dummit &amp; Foote's "Abstract Algebra", 3.1.36 asks the following (which is exactly the same as exercise 5 in ):     Prove that if $G/Z(G)$ is cyclic, then $G$ is abelian. [If $G/Z(G)$ is cyclic with generator $xZ(G)$, show that every element of $G$ can be written in the form $x^az$ for some $a \in \mathbb{Z}$ and some element $z \in Z(G)$]      The hint is actually the hardest part for me, as the quotient groups are somewhat abstract. But once I have the hint, I can write:  $g, h \in G$ implies that $g = x^{a_1}z_1$ and $h = x^{a_2}z_2$, so \begin{align*}gh &amp;= (x^{a_1}z_1)(x^{a_2}z_2)\\\  &amp;= x^{a_1}x^{a_2}z_1z_2\\\ &amp; = x^{a_1 + a_2}z_2z_1\\\ &amp;= \ldots = (x^{a_2}z_2)(x^{a_1}z_1) = hg. \end{align*} Therefore, $G$ is abelian.  1) Is this right so far?  2) How can I prove the "hint"?
Using natural deduction to prove that $p \implies q \vdash \lnot p \lor q$ Using only rules of natural deduction, I am trying to prove that $$p \implies q \vdash \lnot p \lor q$$ but am having a lot of difficulty. I was able to prove the other direction.   Could anyone show me or point me to the right direction? Why the following example prints "0" and what must change for it to print "1" as I expected ?  #include &lt;iostream&gt; struct base {    virtual const int value() const {       return 0;    }    base() {       std::cout &lt;&lt; value() &lt;&lt; std::endl;    }    virtual ~base() {} };  struct derived : public base {    virtual const int value() const {       return 1;    } };  int main(void) {    derived example; } Left pedal coming loose On my bike, which I just built out of spare parts, the left pedal seems to constantly be coming loose. It seems to come out about 5m every time I ride ~10km.  It's an old cotter pin crank, and the pedal is a left pedal (there's an L stamped into the end of the screw that goes into the crank). The pedal has a crack in one side of the axle compartment, but I don't think that would affect it.  I thought that these pedals had left-hand threads so that the action of pedalling would work to tighten the screw? If so, why would this be happening, and is there anything I can do about it short of replacing the crank and pedal? I'm trying to figure out how to send all information that is normally identified in the /var/adm/message file on a Solaris system to a remote system. I want to count the total number of files in particular directory that ends with ".mp4" extension.  I am getting following command:  ls -F |grep -v / | wc -l   It count all the files in particular directory, but I want the count of files that end with .mp4 extension.  Is there any Ubuntu command for that? Multiplying two dimensional data by a constant I have recently been reading a lot on this site (@Aniko, @Dikran Marsupial, @Erik) and elsewhere about the problem of overfitting occuring with cross validation - (Smialowski et al 2010 Bioinformatics, Hastie, Elements of statistical learning).  The suggestion is that any supervised feature selection (using correlation with class labels) performed outside of the model performance estimation using cross validation (or other model estimating method such as bootstrapping) may result in overfitting.   This seems unintuitive to me - surely if you select a feature set and then evaluate your model using only the selected features using cross validation, then you are getting an unbiased estimate of generalized model performance on those features (this assumes the sample under study are representive of the populatation)?  With this procedure one cannot of course claim an optimal feature set but can one report the performance of the selected feature set on unseen data as valid?  I accept that selecting features based on the entire data set may resuts in some data leakage between test and train sets. But if the feature set is static after initial selection, and no other tuning is being done, surely it is valid to report the cross-validated performance metrics?   In my case I have 56 features and 259 cases and so #cases > #features. The features are derived from sensor data.  Apologies if my question seems derivative but this seems an important point to clarify.  Edit: On implementing feature selection within cross validation on the data set detailed above (thanks to the answers below), I can confirm that selecting features prior to cross-validation in this data set introduced a significant bias. This bias/overfitting was greatest when doing so for a 3-class formulation, compared to as 2-class formulation.  I think the fact that I used stepwise regression for feature selection increased this overfitting; for comparison purposes, on a different but related data set I compared a sequential forward feature selection routine performed prior to cross-validation against results I had previously obtained with feature selection within CV. The results between both methods did not differ dramatically. This may mean that stepwise regression is more prone to overfitting than sequential FS or may be a quirk of this data set. I have a server running with the timezone set to UTC. It seemed like that was generally a good practice (please correct me if I'm wrong).  Anyhow, one of the servers I connect to, in order to scp files, is running on EDT and stores files that I need to copy in the format /path/to/filename/data20120913  I looked at trying to rsync files using something like find's -mtime -1 flag for files modified in the last day, but I didn't have any luck.  I don't mind just using scp to copy the current day's file, but as of right now there is a 4-hour window where running date +%Y%m%d will give a different day on each server and that bugs me a little.  Looking through man date I see that I can have the time output as UTC, but I don't see a way to have it output as another timezone like EDT  I suppose I could also use something like the GNU date extension date -d 20100909 +%s to get the date in seconds from the epoch, apply a manual 4 * 60 * 60 second calculation, and see about rendering that as a date - but then when daylight time kicks in it will still be an hour off.  Is there a simpler way to output the date in a YYYYMMDD format for EDT on a server that is set to UTC ? What's the derivative of: $ \sqrt{x+\sqrt{{x}+\sqrt{x+\cdots}}}$? let $ y=\displaystyle \sqrt{x+\sqrt{{x}+\sqrt{x+\cdots}}}$, i'm really interesting to know how do I find :  $\displaystyle \frac{dy}{dx}$ ?.  Note: I have used the definition of derivative of the square root function but i don't succed .  Thank you for any help Help for Mathematics High school I'm looking for mathematics books, courses + exercises for high school level. My daughter was high school certificated but she has stopped her studies for 5 years. She would like to be involved in new educational program but she needs to restore her mathematic capabilities.  Could you help me to identify which mathematics books can help her?  Thanks a lot for your reply,  FR. Add a binary to my path I have an executable. I want to execute the executable in terminal with name only like other commands.  I can put my executable in /usr/local/bin or I could add its PATH to ~/.bashrc. Both will work.  What is better?  Is there  any difference? Sphere inside a conical vessel touches the lid Is there a keyboard shortcut to "unselect" in Windows Explorer? Load R package from character string Why is paper more frangible when it is wet? My four-year-old daughter asked me why paper tends to fall apart when wet, and I wasn't sure.  I speculated that the water lubricates the paper fibers so that they can untangle and separate more easily, but I really wasn't sure. Chess Piece Combinations I came across a question yesterday about combinations, and I wanted to know what the correct answer was. The question states as follows:  There are 8 spaces that are alternately black and white. There is one king, one queen, two identical rooks, two identical bishops, and two identical knights. The king needs to be surrounded by the two rooks. Then, the two bishops can be put on any of the remaining spaces as long as one is on a black space and the other is on a white space. Finally, the queen and the knights can be placed anywhere in the remaining spaces in relation to the other pieces. How many possible arrangements are there considering these conditions? Nullable type issue with ?: Conditional Operator Could someone explain why this works in C#.NET 2.0:      Nullable&lt;DateTime&gt; foo;     if (true)         foo = null;     else         foo = new DateTime(0);   ...but this doesn't:      Nullable&lt;DateTime&gt; foo;     foo = true ? null : new DateTime(0);   The latter form gives me an compile error "Type of conditional expression cannot be determined because there is no implicit conversion between '&lt;null&gt;' and 'System.DateTime'."  Not that I can't use the former, but the second style is more consistent with the rest of my code. I have a list of dictionaries and want each item to be sorted by a specific value. Take into consideration the list: [{'name':'Homer', 'age':39}, {'name':'Bart', 'age':10}]  When sorted by name, it should become: [{'name':'Bart', 'age':10}, {'name':'Homer', 'age':39}] prove that $\mathbb{Q}^n$is dense subset of $\mathbb{R^n}$ Referee says the proof is wrong, but it is not, what to do? About 3 months ago, my co-author and I submitted a mathematical paper to a journal. Yesterday, we got the editor's response and the paper has been rejected. The reason is that the referee thinks that our results are wrong or trivial.  However, we carefully read the referee's report and we conclude that the "errors" pointed out by the referee are not errors at all and that our proofs are correct. In all modesty, we think that the referee understood very poorly even the statements of our theorems.  Personally, this is the first time that such a situation occurs to me. I already have four papers published in such journal and the reports I got from the referees have been always really accurate.  What do you think is the best thing to do? Forget about and try another journal? Write an e-mail to the editor explaining why we disagree with the decision? Other ideas?  EDIT: Thank you very much for all your answers. At the end, we submitted our paper to another journal with only some slight edits we have already planned to do independently by the referee's report.