The free group of rank $n$ mod its commutator subgroup is isomorphic to the free abelian group of rank $n$
Abelianization of free group is the free abelian group
Why can Arwen decide her mortality? How to normalize my db schema for a quiz application? I am putting together a quiz application.   (I'm calling each quiz a "challenge")  I have a "challenge" table and a "challenge_question" table which is pretty straightforward.  challenge table: id name num_questions (I want to randomize and only display a certain number of Qs per each challenge)  challenge_question table: id challenge_id question (text of the question) required (some questions will be required to be displayed per each challenge) question_type (here I indicate what type of question: Mult. choice, T/F, Ordering, or Fill-in-the-blank)   Where I'm having trouble in normalization is with the types of question options:  Multiple choice, true/false, ordering, and fill-in-the-blank. I wonder, should I have a separate table for each type of challenge-question-option? Or is it possible to combine these into one somehow..  If I combined all possible options into a single table, it might look be something like this  challenge_options table: id challenge_question_id option (text field, can be used for both multiple choice and for fill-in-the-blank style questions) is_true (boolean field for true/false questions) is_correct (boolean, used to determine multiple choice answer) ordering_option_1  ordering_option_2 ordering_option_3 ordering_option_4 (these ordering fields would store the correct sequence of an ordering style of question)   This really strikes me as less than ideal.  Actually, I don't see how the ordering_options would work out correctly at all.  Wouldn't I need to have each of those be a separate option?  Surely there must be a better way.  Any suggestions/criticisms or alternate approaches? Testing random variate generation algorithms Which methods are used for testing random variate generation algorithms? Preprocess categorical variables with many values I have a dataset that consists of only categorical variables and a target variable. I want to predict the (binary) target variable with the categorical variables.  I am trying to do this in Python and sklearn.  The categorical variables have many different values. I was thinking to create dummy variables for each value in all the categorical variables. However, in the end this results in a sparse data set with thousands of variables.  How would you go about to reduce the number of dummy variables? Would you use Chi2 to select useful features?  Or would you not make dummy variables at all? Evaluate the limit of $\lim_{n\to\infty}\frac{1}{n^2}\left(\frac{2}{1}+\frac{9}{2}+\frac{64}{9}+\cdots+\frac{(n+1)^{n}}{n^{n-1}}\right)$ can't get placeholder attribute to work in IE using iframe Why shouldn't I care what model/make/year my bicycle is? How to make a logo where a letter is made of particles of varying sizes? How can I make a logo in Adobe Illustrator like the AntiParticle logo below, from ?   It is a large number circles of different, varying sizes, which form the shape of a text character. Stirling's Approximation Proof I was searching for the reason why Stirling's Approximation holds true. I found the website  which apparently shows why this is the case. Does this part of the equation make sense in the proof? If so why?  $$\int_{1}^{n}\ln(x)dx\approx\sum_{k=1}^{n}\ln(k)$$ Why is the picture not inserted? How do you unit test private methods? Label and caption without float I’d like to have an environment that works much like a float, except that it doesn’t float.  That is, in fact, I’d like to be able to define an environment that has a label and a caption.  I explicitly do not want to use a floating environment, though. For one thing, floats do many things fancy which I just don’t need, nor want (such as spacing) and for another, floats have their share of problems (for example consider the problem of ).  Is there a ready-made way to achieve this? Of course, I could always write my own based on the result of the other answer, i.e. harnessing \refstepcounter. But I’m not sure how to use that, documentation is scarce, and I’d prefer an existing solution. I deleted an answer which had no upvotes or downvotes, but after some time I saw that my reputation was -2 and the reason is linked to my search engine.    I want to know why this happened and what "linked to my serach engine" means? Transparent textured planes produce black artifacts Stack Exchange OpenID credentials not removed on logout, allowing insecure re-login What is the difference between a HMAC and a hash of data? How do I get a list of obsolete packages? I'm trying to figure out a way to get a list of the packages that are no longer available in the repositories that I have enabled. This workstation has been through quite a few versions of Ubuntu and has had many 3rd party repositories added and removed. I'd like to get a list of software that I have from these removed repositories, so I can clean it up or add back the appropriate repositories. How to define orbital angular momentum in other than three dimensions? In classical mechanics with 3 space dimensions the orbital angular momentum is defined as  $$\mathbf{L} = \mathbf{r} \times \mathbf{p}.$$  In relativistic mechanics we have the 4-vectors $x^{\mu}$ and $p^{\mu}$, but the cross product in only defined for 3 dimensions. So how to define orbital angular momentum e.g. in special relativity in terms of 4-vectors? Or more generally in $d$ dimensions? Let $A$ and $B$ be $n\times n$ matrices such that $ABAB=0_{n\times n}$.  Can we conclude that  $BABA$ must be  $0_{n\times n}$?   I fail to give a counterexample when $n=2$， so I guess the answer is "yes" How to pull email address and password from text file with CMD? I have a text file named file.txt Which contains,  Combo: testing123mail@gmail.com:password As Combo:  abcdefghijklmnopqrstuvwxyz@gmail.com:ajfbdf some text here some more text here As Combo: hadvsfhia@ahfjbadj.com:password@1 some random text here iloveu@gmail.com:passypassyword123 goldenmammy@hotmail.com:youtube123   I am running the below code, to filter only mail:pass from the above text file file.txt  for /f "tokens=3" %%a in ('type file.txt^|find "As Combo:"') do &gt;&gt;mail.txt echo %%a for /f "tokens=2" %%b in ('type file.txt^|find "Combo:"') do &gt;&gt;mail.txt echo %%b for /f "tokens=1" %%c in ('type file.txt^|find ":"') do &gt;&gt;mail.txt echo %%c   Expected Output in mail.txt,  abcdefghijklmnopqrstuvwxyz@gmail.com:ajfbdf hadvsfhia@ahfjbadj.com:password@1 testing123mail@gmail.com:password iloveu@gmail.com:passypassyword123 goldenmammy@hotmail.com:youtube123   But, am getting actual output is,  abcdefghijklmnopqrstuvwxyz@gmail.com:ajfbdf hadvsfhia@ahfjbadj.com:password@1 testing123mail@gmail.com:password Combo: Combo: Combo: As As iloveu@gmail.com:passypassyword123 goldenmammy@hotmail.com:youtube123   Note: The mail id and passwords will be differ in different text file. Above given mail id and passwords are typed by me randomly for example.  Kindly fix my code and help to get expected output. Thanks in advance.