Proving $A_5$ Has No Subgroup of Order 30
How to prove that $A_5$ has no subgroup of order 30?
I was reviewing the  recently, and it occurred to me that one could allow for only scalar multiplication by the integers and still satisfy all of the requirements of a vector space.   Take for example the set of all ordered pairs of integers. Allow for scalar multiplication over the integers and componentwise vector addition as usual. It seems to me that this is a perfectly well-defined vector space.  The integers do not form a , which begs the question: Is there any reason that the "field" over which a vector space is defined must be a mathematical Field? If so, what is wrong with the vector field I attempted to define above? If not, what are the requirements for the scalars? (For instance, do they have to be a  - Abelian or otherwise?) Improving "demonstrate a minimal understanding" close reason How to change charset of my website to utf8? Upgrade to version newer than repositories I have  about how to upgrade ghostscript to a version that is available on 15.04 but not on 14.04LTS (which I have). On this site there are numerous almost identical question regarding different packages, a few examples include:         So I thought it would be very useful in general to know: is there  a general, automated way to upgrade a package, including all its dependencies, to a version that is newer than Ubuntu's repos? Is this more easily done if the newer version exists in more recent Ubuntu releases? Why isn't Force Speed used more often? Is there a proof from the first principle that the Lagrangian L = T - V? Is there a proof from the first principle that for the Lagrangian $L$,    $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$  in classical mechanics? Assume that Cartesian coordinates are used. Among the combinations, $L = T - nV$, only $n=1$ works. Is there a fundamental reason for it?  On the other hand, the variational principle used in deriving the equations of motion, Euler-Lagrange equation, is general enough (can be used to to find the optimum of any parametrized integral) and does not specify the form of Lagrangian. I appreciate for anyone who gives the  answer, and if possible, the primary source (who published the answer first in the literature).      Notes added on Sept 22: - Both answers are correct as far as I can find. Both answerers were not sure about what I meant by the term I used: 'first principle'. I like to elaborate what I was thinking, not meant to be condescending or anything near to that. Please have a little understanding if the words I use are not well-thought of. - We do science by collecting facts, forming empirical laws, building a theory which generalizes the laws, then we go back to the lab and find if the generalization part can stand up to the verification. Newton's laws are close to the end of empirical laws, meaning that they are easily verified in the lab. These laws are not limited to gravity, but are used mostly under the condition of gravity. When we generalize and express them in Lagrangian or Hamiltonian, they can be used where Newton's laws cannot, for example, on electromagnetism, or any other forces unknown to us. Lagrangian or Hamiltonian and the derived equations of motion are generalizations and more on the theory side, relatively speaking; at least those are a little more theoretical than Newton's laws. We still go to lab to verify these generalizations, but it's somewhat harder to do so, like we have to use Large Hadron Collider. - But here is a new problem, as @Jerry Schirmer pointed out in his comment and I agreed. Lagrangian is great tool if we know its expression. If we don't, then we are at lost. Lagrangian is almost as useless as Newton's laws for a new mysterious force. It's almost as useless but not quite, because we can try and error. We have much better luck to try and error on Lagrangian than on equations of motion. - Oh, variational principle is a 'first principle' in my mind and is used to derive Euler-Lagrange equation. But variational principle does not give a clue about the explicit expression of Lagrangian. This is the point I'm driving at. This is why I'm looking for help, say, in Physics SE. If someone knew the reason why n=1 in L=T-nV, then we could use this reasoning to find out about a mysterious force. It looks like that someone is in the future. Is $\lim\limits_{n\rightarrow\infty} n$ comparable to $\aleph_0$? For example, can we say: $\infty=\lim\limits_{n\rightarrow\infty} n &lt; \aleph_0$?  These are two different types of structures. The limit being like the length, extension, or just generic magnitude and the other being cardinality of a set. Can we compare magnitude to cardinality?  Intuitively, we can reach $\aleph_0$ by counting the natural numbers on the number line and in the process will be approaching $\infty$. Which leads me to believe $\infty\leq\aleph_0$. But I can't see why it should be a strict inequality. I feel like they should be of equal magnitude.  I saw on a recent comment that $2^\infty=\infty$, but are those infinities really the same? It seems not to me. Of course we (usually) have that  $2^{\aleph_0}=\aleph_1$ where ${\aleph_0}$ and $\aleph_1$ are clearly two very distinct infinities, countable vs uncountable at least. Maybe one might argue that as far as the concept of magnitude is concerned, all infinities are "equal". Prove directly from the definition of the Ito's integral I am trying to solve the exercises from the book Stochastic differential equations -An Introduction with applications by Bernt Oksendal and I am stuck on 1 question.  Prove directly from the definition of Ito's integral that   $$\int_0^t B_s^2 dB_s=\frac{1}{3}B_t^3-\int_0^t B_s ds$$  I have tried for a couple of hours but I simply cannot prove it . I would be grateful if somebody could help me out , even a hint would be very helpful Is there a built-in function to print all the current properties and values of an object? How to add Windows to GRUB I have my encrypted Archlinux installed on my SSD. Now I have one HDD with Windows installed. But in my GRUB just Archlinux is shown (no Wonder, because I just installed Windows on the other HDD, GRUB doesn't know of that by now.).  My Question is, how do I properly add my Windows-HDD to GRUB, so that I can choose which to boot when starting?  I am a little bit afraid of destroying my GRUB so I can't boot my encrypted Archlinux anymore.  Thanks! Why does this CSS margin-top style not work? New breaker and panel: What's the brown goo where they connect? 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