In the group $S_n$ consisting of the set of all bijections from $\{1,2,...,n\}$ to $\{1,2,...,n\}$ with a binary operation $◦$ denoting composition of functions. I am asked to find cyclic subgroups of $S_4$ with 2,3, and 4 elements. For a cyclic subgroup with 2 elements I have $\{(1,3),(2,4),(3,1),(4,2)\}$, but am looking for help in finding cyclic subgroups of larger sizes.
I would like to count subgroups of each order (2, 3, 4, 6, 8, 12) of $S_4$, and, hopefully, convince others that I counted them correctly.  In order to do this by hand in the term exam, I need a clever way to do this because there can be as many subgroups of a group of order 24 as $2^{23}$.  Do you know how to do this?   (I would be most grateful if you could tell me what part of the answer to the old question answers my question before voting to close.)
How do I show that $\ln(x) = \lim_{n\to\infty} n (x^{1/n} - 1)$?  I ran into this identity on . I haven't been able to find any proof online and my efforts to get from $\ln(x) := \int_1^x \frac{\mathrm dt}t$ to that limit have been a failure. The first and (the) second chapter(s) are difficult - which is correct? I'm a little confused about the following sentences. Could you please tell me which one(s) is/are correct?        The first and the second chapter are difficult.   The first and the second chapters are difficult.   The first and second chapters are difficult.   The first and second chapter are difficult.      I believe #1 &amp; #3 are correct, while #2 &amp; #4 are incorrect.  But when seeing the answers to the Quora question , I wonder why they say #2 is correct. Installing PHP 5.6 on Xenial (16.04) I need to use php5 (5.6) because I'm working on a Drupal 7 project which doesn't support php7. The problem is I recently updated to ubuntu 16.04 Xenial which seems to .   I tried to install it from , however when It try to install it with apt I always get similar errors.  $ sudo apt-get install php5-cli Cependant les paquets suivants le remplacent :   php7.0-cli:i386 php7.0-cli`   Which means it's obsolete. How can I install PHP 5.6 on Xenial? Is selenium webdriver capable of database testing? After reading so many answers and forums it makes me confused. Some people say: Selenium webdriver is a browser automation tool and it has nothing to do with the database. And some other showing JARs and code that how we can connect selenium to the database and we can test database using selenium webdriver. So I would really like to know if we can do database testing using selenium? If yes then which databases tool it supports like Mysql, SQL server, Oracle... And if selenium is not suitable for database testing then please recommend to me the best tool for database testing. Im trying to understand the solution about this, and all the other solutions are.. a bit weird for me and dont answer all my questions!  Reference :   Im using the answer there as reference, but my understanding is a bit different!     Find the error in the following proof that all horses are the same   color. CLAIM: In any set of h horses, all horses are the same color.      PROOF: By induction on h.      Basis: For h = 1. In any set containing just one horse, all horses   clearly are the same color. Induction step: For k > I assume that the   claim is true for h = k and prove that it is true for h = k + 1. Take   any set H of k + 1 horses. We show that all the horses in this set are   the same color. Remove one horse from this set to obtain the set H1   with just k horses. By the induction hypothesis, all the horses in H,   are the same color. Now replace the removed horse and remove a   different one to obtain the set H2 . By the same argument, all the   horses in H2 are the same color. Therefore all the horses in H must be   the same color, and the proof is complete.   Here's what i have come up with :  Suppose a set of 1 horse. $H_a = { horse_{1} }$  Obviously, all the horses in set $H_a$ are the same, since theres just 1 horse.  So using that as the Basis of the induction we can proceed and try to prove its true for $N$ horses.  -- I dont fully understand the part below   As an example : set $H_0$ :   $H_0 = { horse_{1}, horse_{2}, horse_{3}... horse_{n} }$  In my understanding, its because, I can make a subset  formed by 1 horse :  $K_1  = {horse_{1}}$ $K_2  = {horse_{2}}$ ...  I don't get it. I clearly know that this proof is a lie.  And for me, its because you cant claim theres a relation on a set with size $&gt;= 2$ , since :  $H_1 = { horse_{1}, horse_{2}}$  Theres nothing that says $horse_{1}$ equals $horse_{2}$. The basis was for one horse. But that alone is not a proof.  So what should I do ? 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What is the difference between:  public void myFunction(ref MyClass someClass)   and  public void myFunction(out MyClass someClass)   Which should I use and why? Rice with worm and possibly their eggs I bought a bag of brown rice from a store. After having opened and used for a while, I found there were moths flying in the house. At the same time, I found worms in the rice. I suspect that the moths are from the worms(rice-size, white body, dark head). I guess but am not sure if they are called rice moths. There were also many little sand-like things in the rice bag, are they the eggs of the rice moths?  Will eating this infested rice pose a health risk? Async call generates " Error: Can't wait without a "fiber", even with _wrapAsync Is God either immoral or not omnipotent? The usual (Christian) justification for suffering/evil in the world created by a benevolent God is freedom of the will. However, the more interesting question is not about the source of evil (which the free will may very well be) but why God chose to create the world such as it is. Why not create a world with freedom of the will, but without suffering? Even assuming that God is bound by logic such combination doesn't seem to be logically impossible: creatures certainly can do good of their own free will. There are two alternatives.  1) God is powerful enough for all practical intents and purposes, enough to be called God, but not omnipotent. He created only the best of all possible worlds (Leibniz). This position is perhaps the most defensible philosophically, it also avoids logical problems with , etc., but it does diminish deity's status and invites unpleasant questions like what limits God (logic only, inert matter, outer chaos, other beings), is there someone/thing more powerful, etc.    Interestingly enough, the original Christian theodicy wasn't free will. St.Augustine says poetically "the canvas of creation requires both the black and the white paint". "Requires" here can be interpreted differently,  is to explain the black by the "just punishment" for the original sin, which brings us to Adam and Eve, and back to the free will. But if God is still omnipotent and hence omniscient creating creatures he knew would sin, and then punishing them for it is hardly an improvement.   2) However, Augustine can be read more straightforwardly. Evil is "required" for the perfection of the world just like black paint is required for the perfection of a painting (evil can be interpreted as lack of good rather than something onto itself but that is beside the point). God is immoral, and he is beyond our conception of morality, his notion of perfection is his own. This is much bolder, and similar to Plotinus's position on the existence of the One, the One can not exist because it is "prior to all existence". But it does evoke the unpleasant taste of "ends justify the means". It is one thing to use "God is beyond our logic and comprehension" to get rid of unliftable stones, it is quite another to convince people to worship an immoral perfectionist.  Formally, there is another logical possibility of asserting both morality and omnipotence, but defending the morality of creating a world full of suffering, for a higher purpose perhaps. For example, the suffering only appears bad to us, not to an absolute being in its wisdom. But this simply redefines morality and is only different in phraseology from the Augustine's position. The question is asked in human terms and has to be answered in kind. One is free to argue that God is beyond human logic and/or morality, but it still leaves him immoral as commonly understood.   There does not appear to be a way to square this circle, one has to accept Leibniz's or Augustine's alternative (or both) and deal with the downsides. 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Tracking down where disk space has gone on Linux? When administering Linux systems I often find myself struggling to track down the culprit after a partition goes full. I normally use du / | sort -nr but on a large filesystem this takes a long time before any results are returned.  Also, this is usually successful in highlighting the worst offender but I've often found myself resorting to du without the sort in more subtle cases and then had to trawl through the output.  I'd prefer a command line solution which relies on standard Linux commands since I have to administer quite a few systems and installing new software is a hassle (especially when out of disk space!) How to disable or uninstall Microsoft Edge in Windows 10