Minimal non-cyclic groups other than the Klein four groups everyone. I am afraid that my question is too trivial. But here it is. The Klein four group is the first counterexample to the the statement: "If all proper subgroups of a group are cyclic, then the group is cyclic." I am looking for other examples, if any. Are there?  Thanks in advance.
Is there some sort of classification of all minimal non-cyclic groups? Does there exist some sort of classification of all minimal non-cyclic groups (non-cyclic groups, such that all their proper subgroups are cyclic)  I know the following classes of such groups:  1) $C_p × C_p$, where $p$ is a prime  2) $Q_8$  3) $\langle a,b | a^p = b^{q^m} = 1, b^{−1}ab = a^{r}\rangle$, where $p$ and $q$ are distinct primes and $r ≡ 1 \pmod q$, $r^q ≡1 \pmod p$.  (These three classes completely cover the case, when our group is finite: )  4)$C_{p^{\infty}}$, where $p$ is a prime  5)$(\{ \frac{n}{p^m}| m, n \in \mathbb{Z} \}, +)$, where $p$ is a prime  (These two classes completely cover the case, when our group is infinite abelian: )  6)Infinite non-abelian groups, such that all their nontrivial proper subgroups are isomorphic to $C_{p}$ for a fixed prime $p$ (Tarski monster groups)  7)Infinite non-abelian groups, such that all their nontrivial proper subgroups are isomorphic to $C_{\infty}$ ().  However, I do not know, whether there exists anything, that does not fall into these classes. I only know, that if such groups exist, they have to be infinite non-abelian.
Consider the function : $$f: \mathbb{R}^2 \rightarrow \mathbb{R} , (x,y) \mapsto   \begin{cases}         0 &amp; \text{for } (x,y)=(0,0) \\    \frac{x^3}{x^2+y^2}       &amp; \text{for } (x,y) \neq (0,0)   \end{cases} $$  Show that $f$ not differentiable at $(0,0)$ but all directional derivatives exist.  I don't know how to tackle this problem. Can someone give some hints or solve the problem? Thanks :) Finding the derivatives of sin(x) and cos(x) We all know that the following (hopefully):  $$\sin(x)=\sum^{\infty}_{n=0}(-1)^n \frac{x^{2n+1}}{(2n+1)!}\ , \ x\in \mathbb{R}$$  $$\cos(x)=\sum^{\infty}_{n=0}(-1)^n \frac{x^{2n}}{(2n)!}\ , \ x\in \mathbb{R}$$  But how do we find the derivates of $\sin(x)$ and $\cos(x)$ by using the definition of a derivative and the those definition above?  Like I should start by doing:  $$\lim_{h\to\infty}\frac{\sin(x+h)-\sin(x)}{h}$$  But after that no clue at all.  Help appreciated! Where does Drupal store its content? I have been currently learning Drupal while trying to duplicate a Drupal application into CakePHP.  I have been looking at the  database tables to know where data is being stored so I can safely migrate any data or study how it is working.  I have researched and have come across questions like:     So far, I have only seen the title in my Drupal (7) installation.  What I have seen in my node revisions table for a recipe content type, which includes fields like, field_ingredients, field_instructions, field_description, field_preptime, field_cookingtime, just the title of the recipe. This is a sample record on my node_revision table.  nid   |  vid  |  title         |  log   | timestamp  | status | comment | promote  | sticky ------------------------------------------------------------------------------------------- 1     |  1    |  Awesome Sauce |        | 1369998600 | 1      | 2       | 1        | 0   As you can see I can only see the title. Where are the other field contents? What defines the upper limit for speed of Subspace Communications? mentions there to be a definite upper limit as to the speed of Subspace Communications. Is the reason for this ever touched on in any of the Star Trek media (Canon or not) or is it "speed of plot"? This  would suggest speed of plot but I'm curious what would provide an upper limit if light doesn't? Launch application on a specific monitor I have Ubuntu 14.04 and two monitors. When I open applications I need some of them to be on the primary monitor and others On the secondary. At the moment, apps always open in the primary monitor.   I have dash icons only on one (left) display, so I can't launch apps using the cursor or active window setting from Compiz. Is there a way to easily turn on/off showing the boot messages (loading the services) when Ubuntu starts? Is it something in Grub2?  I am running 10.04. Can't move/snap Nautilus window to occupy left or right half of the 1366x768 screen - is it just me? I'm running Ubuntu 18.04 LTS with GNOME Shell. Nautilus does not want to occupy 50% (left or right) of the screen sized at 1366x768.    Other applications (such as GNOME Terminal, Eye of GNOME, Gedit) are normally snapping to the left/right half of the screen.  But Nautilus can be resized to the width lower than half of screen (I can get 650 px, which is smaller than 1366/2 = 683 px).  The first usable screen resolution seems to be 1400x1050, then goes 1440x900. But large amount of current devices .  Is it just me?     Updates: 1. standard shortcut Super+← or Super+→ do not work too. 2. I reported  which has linked upstream . 3. It does not work on GNOME FlashBack session too. Difference between might and could The title pretty much says it all: how can I type Unicode characters - specifically the em-dash —? WARNING DO NOT ATTEMPT TO RUN THIS ON A PRODUCTION MACHINE   In reading the  I generally follow what's going on with the following code:  :(){ :|:&amp; };:   excerpt of description     The following fork bomb was presented as art in 2002;   its exact origin is unknown, but it existed on Usenet prior to 2002.   The bomb is executed by pasting the following 13 characters into a    shell such as  or . It operates by defining   a function called ':', which calls itself twice, once in the   foreground and once in the background.   However the last bit isn't entirely clear to me. I see the function definition:  :(){ ... }   But what else is going on?  Also do other shells such as ksh, csh, and tcsh also suffer the same fate of being able to construct something similar? What are tag synonyms and merged tags? How do they work? How do you compare two version Strings in Java? Is there a standard idiom for comparing version numbers?  I can't just use a straight String compareTo because I don't know yet what the maximum number of point releases there will be.  I need to compare the versions and have the following hold true:  1.0 &lt; 1.1 1.0.1 &lt; 1.1 1.9 &lt; 1.10 There appears to be a giant security vulnerability with Android that seems to basically affect all phones.  wrote:  The vast majority of Android phones can be hacked by sending them a specially crafted multimedia message (MMS), a security researcher [Joshua Drake] has found. ... Drake found multiple vulnerabilities in a core Android component called Stagefright that’s used to process, play and record multimedia files. Some of the flaws allow for remote code execution and can be triggered when receiving an MMS message, downloading a specially crafted video file through the browser or opening a Web page with embedded multimedia content. ... The MMS attack vector is the scariest of all because it doesn’t require any interaction from the user; the phone just needs to receive a malicious message. For example, the attacker could send the malicious MMS when the victim is sleeping and the phone’s ringer is silenced, Drake said. After exploitation the message can be deleted, so the victim will never even know that his phone was hacked, he said.  What can a regular user do to mitigate the issue? Disable Google Hangouts? Find all entire functions $f(z)$ such that $|f(z)|=1$ for $|z|=1$ AMD driver does not work on kernel 3.19.0-26 Second law of Newton for variable mass systems Frequently I see the expression  $$F = \frac{dp}{dt} = \frac{d}{dt}(mv) = \frac{dm}{dt}v + ma,$$ which can be applied to variable mass systems.  But I'm wondering if this derivation is correct, because the second law of Newton is formulated for point masses.   Furthermore if i change the inertial frame of reference, only $v$ on the right side of the formula $F = \frac{dm}{dt}v+ma$ will change, meaning that $F$ would be dependent of the frame of reference, which (according to me) can't be true.   I realize there exists a formula for varying mass systems, that looks quite familiar to this one, but isn't exactly the same, because the $v$ on the right side is there the relative velocity of mass expulsed/accreted. The derivation of that formula is also rather different from this one.   So my question is: is this formula, that I frequently encounter in syllabi and books, correct? Where lies my mistake. Does it matter what I choose for serialVersionUID when extending Serializable classes in Java? A proof for the inequality $\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2} $ for all $t \geq 2$ I'm struggling with proving the following inequality: $$\sum_{i=0}^{t-2}{\frac{1}{t+3i}} \leq \frac{1}{2}$$  for all $t \geq 2$. I think it is monotonic non-increasing in $t$, which would suffice.  Thanks in advance! Does deleting iCloud photos affect them on devices? If I delete a photo from my iCloud will it also be removed from my devices that are connected to my account? How many fixed points in a permutation I am taking a probability course and we have been of course learning the classic problems of men with hats or variations of that in some other form. The problems being of the type if $n$ men have $n$ hats and mix them all together, whats the probability $0, 1, 2,$ or $k$ men choose their hat.   But from what I can gather is that they are all problems of the same type, for any given random permutation of $k$ objects how many fixed points are there. My professor taught us to solve this using conditional probabilities (conditioning on whether the first person matched and then generating a system of equations) and using the inclusion-exclusion principle. But is there any other way to reason this?