Can anyone provide with an explanation of why the group $A_4$, which is the group formed by the set of even permutations of $S_4$ under the operation of composition of functions, can not have an order of $6$? I know Lagrange's Theorem tells us that the orders of possible subgroups of $A_4$ are $1,2,3,4,6,12$, and I can find a subgroup a subgroup of all of the orders listed except $6$, and I'm pretty sure there is not one, but cannot come up with a solid explanation as to why this cannot happen?  Thanks for the help!
Show that  the  alternating  group  $A_4$  of all even permutations of $S_4$ does not contain a  subgroup of order $6$.   For me am thinking to write all elements of $A_4$ and trying to find every cyclic subgroup generated by each element of $A_4$, then I have to check whether there exist such a subgroup or not! This is a long procedure for me, I ask if there is a short way to do this.
How would Voldemort restore himself from a horcrux? If Voldemort's body dies, what process would he use to restore himself from a horcrux? His plans seem sloppy for the pains he had taken. For example, why create horcruxes from pieces of jewellery when they dont have the agency to seek out a new body by themselves?  He needed Quirrel, a person with unlikely ambitions to seek him out to gain an useful body. When that body was gone, he needed Barty Crouch, someone with questionable agency (he could be imprisoned) to transport Harry and draw his blood.  Nagini and the diary seems the only horcruxes with agency, the others are mostly inert.  What plans were ever taken to restore Voldemort from his horcruxes? Should we use plural or singular for a fraction of a mile? I have seen people say both 0.25 mile and 0.25 miles. Should we use plural or singular for a fraction of a mile? How can I remove a key from a Python dictionary? I read about the new fancy progress bar which was added to apt 1.0 in  and how to enable it . However on running sudo apt-get update, I did not see the fancy progress bars. On reading the second link again, the solution explicitly requires you to use sudo apt update.  Hence my question is the difference between apt and apt-get. I thought they were one and the same. Do I have to code each case of this Grid full of plots separately? Prove $e^{x+y}=e^{x}e^{y}$ by using Exponential Series In order to show $e^{x+y}=e^{x}e^{y}$ by using Exponential Series, I got the following:   $$e^{x}e^{y}=\Big(\sum_{n=0}^{\infty}{x^n \over n!}\Big)\cdot \Big(\sum_{n=0}^{\infty}{y^n \over n!}\Big)=\sum_{n=0}^{\infty}\sum_{k=0}^n{x^ky^n \over {k!n!}}$$  But, where should I go next to get $e^{x+y}=\sum_{n=0}^{\infty}{(x+y)^n \over n!}$.  Thanks in advance. Brand new disk brakes are barely stopping the bike There are no compact minimal surfaces Is everything greater than None? Looking for a Simple Argument for "Integral Curve Starting at A Singular Point is Constant" Let $U$ be an open subset of $\mathbf R^n$ and $V:U\to\mathbf R^n$ be a differentiable vector field on $U$. Let $\mathbf p\in U$ be a singular point of $V$, that is, $V(\mathbf p)=\mathbf 0$.     Then the only integral curve which starts at $\mathbf p$ is the constant curve.   I know that one could simply use the theorem of "uniqueness of integral curves" and in fact adapt the same proof for this particular case.  But is anybody aware of a simple argument for this? Where is the line drawn with domain names which include a trademark? A search on google for "iphone developer" turns up loads of websites which have "iphone" in them, a trademarked name by Apple.  So I'm led to believe that having a domain such as iphonedeveloper.com is ok? Well, you're still using Apple's trademark, but it would be hard to brand yourself otherwise. You're an IPHONE DEVELOPER... right?  Well, what if I want to provide a website where users pay to get a list of the best offers from Ebay? I might have a domain like ebaydeals.com (I don't... i'm just speculating!).  Now I've heard that places like Ebay are really hot on the trigger and fire out emails to people who register domains like that straight away. But whats the difference?  In both cases I'm making money from the trademark, effectively, so is it just down to how lenient the company who owns the trademark is? Or are there rules? Is there a specific "line" you don't cross?  Thanks Difference between "been to" and "been in" $n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $n \mid \frac{a^{n}-b^{n}}{a-b}$ How does one prove that if $n \mid (a^{n}-b^{n}) \ \Longrightarrow$ $ \displaystyle n \mid \frac{a^{n}-b^{n}}{a-b}$ where $a,b, n \in \mathbb{N}$.  What i thought of is to consider $$(a-b)^{n} \equiv a^{n} + (-1)^{n}b^{n} \ (\text{mod} \ n)$$ and if we suppose that $n$ is odd then we have, $$(a-b)^{n} \equiv a^{n} -b^{n} \ (\text{mod} \ n)$$ and since $n \mid (a^{n} - b^{n})$ we have $$(a-b)^{n} \equiv 0 \ (\text{mod} \ n) $$  I think i am far away from the conclusion of the problem, but this is what i could work on regarding the problem. What can I do with .chk files in FOUND.000 folders? Just discovered I have gotten a FOUND.000 folder on my usb drive. I guess from running chkdsk once. It has three files in it:   FILE0000.CHK FILE0001.CHK FILE0002.CHK   What exactly can I do with these files? What are my options? Are they lost files? Are they garbage? Can I use them for anything at all? Or will I just have to delete them? Warning: Total photoshop newb!  I recently got a cool image of my fiancee and I holding hands and showing off her engagment ring. The backdrop is the beach:    I tried to trace an image using a combo of the laso and magic wand tools to select my target. Then I used that to pull out the target as another layer.  I then pasted in a background gradient.     Ended up with a really lame photoshoped looking picture. Notice the white lines where I didn't get the arms close enough and the choppy spots where I got too close.    So what's my main problem here?   Is their a way to bet a better cutout? Is their a blending tool that will make the background look more natural? both   I tried a dozen or so tutorials I found in google searches but they just don't blend these types of images together in any of them. Thanks in advance. What's the best way to take a segment out of a text file? Is there a RSS to the 'Stack Exchange Inbox'? How about to Activity? Just wondering if I'm missing something here. Also, I couldn't find any previous question related to this. How to convert a single char into an int For sets $A,B$, let $|A|\leq^*|B|$ say that there exists an onto map $f:B\rightarrow A$ or $A=\emptyset$.   My question is, is $$\forall A,B(|A|\leq^*|B|\longrightarrow |A|\leq|B|)$$ equivalent to the axiom of choice?  Thanks What are some good books for learning the concepts of Kinematics, Newton laws, 2D Motion of Object etc.?