Let $G$ be a commutative group of odd order.  Show that the map $f(a)=a^2$ is an isomorphism of groups from G to itself.  $f(ab)=(ab)^2=a^2b^2=f(a)f(b)$ so clearly its a homomorphism.  Now suppose $f(a)=f(b)$.  Then $a^2=b^2$.  Multiply on the left by $a$.  $a^3=ab^2$ and since G is odd, $a^3 =e$.  So $e=ab^2$. Then multiplying by $b$ on the right,  $eb=ab^3$ and $b^3 = 1$ so $a=b$. It is injective.  How do we show it is surjective?
Suppose $G$ has odd order, show the function $f:G\rightarrow G$ defined by $f(x)=x^2$ is injective. This proposition is easily provable if we assume $G$ is Abelian, but I don't know how to start this without the assumption of being Abelian.
Is there a word for colloquial forms of address? For example, "dude," "man," "buddy," "pal," etc, when used to stand in for someone's name.  "Hey, pal, how's it going?"  Is there a word for terms like these?  Or is "colloquialism" as close as we can get? Moving /var, /home to separate partition I am attempting to move some folders (such as /var and /home) to a separate partition after reading this guide:    I was able to move one folder successfully following guide.  However, it doesn't seem to work for multiple folders, and all my folders are dumped into the partition without proper folders.  I would like to mount /var, /home, and /tmp onto the separate partition; can someone guide me on this? Are antiparticles just particle-shaped holes? If particles are simply regions of space where certain quantum fields have non-zero divergence, are anti-particles simply the corresponding regions of opposite divergence?    This seems like the intuitive answer, especially when considering the process of annihilation.  I have heard before that anti-particles are analogous to particles moving through time in reverse, which seems to indicate that an annihilation event is just a point of symmetry in that (arguably, single) particle's history.  This analogy breaks down, however, when it comes to gravitation, since there seems to be no evidence of a negative-mass particle.    So what then is the meaning of an anti-particle, when their symmetry is preserved across certain fields, but not other? Some say that we can connect two points with a continuous curve and a small contractible neighborhood of the path together with the charts of the points can be regarded as the chart. But I don't know how to verify this statement. How to find the sum $$\sum^n_{k=1} (k^2+k+1)k!$$  What I tried as follows :   $$\sum^n_{k=1} (k^2+k+1)k!$$  =$$\sum^n_{k=1} (k^2)k!+ \sum^n_{k=1} (k)k! + \sum^n_{k=1} (1)k!$$  Now we can write $\sum^n_{k=1} (k^2) $ as $$\frac{n(n+1)(2n+1)}{6}$$ but what to do with $k!$ please guide thanks....... How can I sort arrays and data in PHP? My phone was just stolen an hour or so ago. Are there any options for tracking? I realize this is a long shot, but my phone was lost or stolen a few hours ago.  At the time it was stolen, I didn't have any specific applications installed to track it or wipe the data. So... my bad. I'm hoping the fact that I have a pattern lock will at least prevent anyone from accessing my data.  My question is, do I have any options for tracking or data wipe after the fact? (I'm more interested in tracking.)  Note: I have read similar questions, but unless I'm mistaken, they all address the question in terms of preparation before losing the phone. I'm asking about fallback options after losing the phone. Is it really impossible to calculate in advance the result of throwing dice? Let $\lambda(A)$ be the Lebesgue measure of $A$. There exists an interval $I$ such that $\lambda(E \cap I) / \lambda(I) < 1-\epsilon$ (Not mentioned in title but $\epsilon$ is a number greater than $0$ and $E$ a Lebesgue measurable subset of $\mathbb{R}^n$.) I know a question equivalent to this one has been asked (). But it was not answered. Also my thinking is a bit different. So I would love to get some help on this problem. So far I've got:  $$\frac{\lambda(E \cap I)}{\lambda(I)} &gt; 1- \epsilon \iff \frac{\lambda(I)-\lambda(E\cap I)}{\lambda(I)} &lt; \epsilon$$  And also, a theorem states that, for all $\epsilon$ there exists a closed set $F\subset E$ such that $\lambda(E\setminus F) &lt; \epsilon$ so it would sufice to prove  that there exists an interval $I$ such that $$\frac{\lambda(I)-\lambda(E\cap I)}{\lambda(I)}&lt;\lambda(E\setminus F).$$ I'm not sure if this helps me much but any help is appreciated. What would you do? At the moment I'm working with the . For this I read the following paper: .  But I have some mathematical problems with this formalism. I don't really see why this is a variation principle since it is opartor-valued.   The action principle is defined by: $\delta S  = G_1 -G_2$, where $S$ is the action operator $S = \int_{t_1}^{t_2} dt ~L$ and $G$ the generator operators of the transformations.  What does it mean that the action principle is operator-valued? For the classical action where $S$ is real we search for a stationary point but for operators? I also have a problem that $\delta S \neq 0$. What interpretation has this additional term $G_1 - G_2$? How can I schedule timer to power on and off device? Prove that $2^{1/2}$ is irrational I'm trying to understand each part of this completed proof that my professor did, here is my interpretation in parentheses, please advise as necessary.  Proof: Assume that $2^{1/2}$ is rational, then $2^{1/2}$ = $p/q$ for some integer $p$ and $q$ with $q \neq 0$ and $\gcd(p,q) = 1$  $$(p/q)^2 = 2 = p^2 / q^2 $$ $$\text{ (What was the purpose of squaring √2 and p/q?)}$$  $$p^2 = 2q^2$$ $$ \text{ (Is this basically just canceling out the q^2 on left side, and multiplying the right side?)}$$  2 | p^2 => (2 divides p^2 since it is a factor)  2 | p => (thus 2 divides p)  p = 2k for some integer k => (definition of even)  (2k)^2 = 4k^2 = 2q^2 => (Why are we squaring 2k, and how does 4k^2 = 2q^2?)   q^2 = 2k^2 => (Canceled out 2 on one side, and dividing on other)  2 | q^2 => (2 divides q^2)  2 | q => (thus 2 divides q)  2 | gcd(p,q) => (2 divides gcd of p and q)  2 | 1 Which is a Contradiction (no clue how I got here) Set the Max depth parameter of both the Reflection and Refraction roll-out to 10. I am not sure whether roll-out should be plural or not in this particular case. Objects with transparent texture not always show transparency in 3dview For some reason objects don't show their transparency all the time.  In this image the circles in red are suposed to be transparent like the ones in green.    When I select the object in the back, all objects in front of it become opaque.    When I select the two objects that are opaque, then they become transparent as they sould be.    The display is in GLSL mode  These are the material and texture properties      I'm not worried about the final render, just the editing view. I would like to see the actual transparency at editing time only, because the scene is to be imported in a third party game engine.  Blender is v2.68   is the blend file Does it make sense to say, "The speed of light varies?"  Some may say right off the bat "Yes, it changes as a wave passes through a different medium."  However, I'd like to say no, because when I hear someone say the speed of light, I always think of the constant $c$ (unless the medium is specified to not be a vacuum, but then it isn't $c$ anymore), not the speed of a particular wave.  To me, it makes more sense to say something like, "The speed of a particular wave varies." What is the correct way to state this in the professional world? And in general, when professional physicists say "the speed of light," are they referring to the constant or the actual speed of the wave? How do I programmatically log in a user with a POST request? Do anyone have a working REST Login on Drupal 8? This is what I tried. POST /user/login HTTP/1.1 Host: 8.d8.local Content-Type: application/x-www-form-urlencoded Accept-Encoding: http://nikhilmohan.in Cache-Control: no-cache  name=test&amp;pass=password&amp;form_id=user_login_form  It returns me HTML instead of JSON. Is there canon info on shelf life of various potions in Harry Potter? Do we know how long potions (in general, or at least specific ones) are potent (no pun intended) in Harry Potter canon?  E.g. do they last a couple of hours? Minutes? Years?   I seem to vaguely remember some comments of Snape's about Wolfesbane potion therfore I think at least some information exists. Using the expression "the same" for a previously mentioned item I frequently use &quot;the same&quot; as in the following sentence.  I need the details of XYZ project. Could you please send me the same as soon as possible?  Is this grammatically correct? I have not been corrected till now, but I have not seen the usage from native speakers very much either. Update: Seems like my example sentence was ambiguous. Hope the following sentence is better.  I need the documents for the meeting. Could you please send me the same as soon as possible? Is there an AmpScript function that can be used in an equivalent manner to sql left, right and mid functions?   Thanks, Mark In the following method definitions, what does the * and ** do for param2?  def foo(param1, *param2): def bar(param1, **param2):