Showing the group of Inner Automorphism are a normal subgroup of the Automorphisms I am trying to show that the group of Inner Automorphisms are a normal subgroup of the Automorphisms of a group G. So far I have this.  Let $\phi_x:G\rightarrow G$ be the inner automorphism defined as $\phi_x(g)=xgx^{-1}  \forall g \in G$. It is clear that $Inn(G)\neq \emptyset$ since $e\in G$ such that $\phi_x(e)=xex^{-1}=e.$  We will show closure and inverses. Observe that $\phi_x\circ \phi_y(g)=\phi_x(ygy^{-1})=xygy^{-1}x^{-1}$.   My question here is for composition of functions, what are you trying to show? Is it just that $\phi_x\circ \phi_y(g)=\phi_{xy}(g)?$.  Now for inverses. (I am trying to follow along with this question here ).  So in my case $\phi_e$ is the identity automorphism. Thus $(\phi_x)^{-1}=\phi_{x^{-1}}$. Thus $(\phi_x)^{-1}\phi_h=\phi_{x^{-1}}\phi_h=\phi_{hx^{-1}}=\phi_e\in Inn(G)$. Thus the Inner automorphisms are a subgroup.  Now to show normality. We need to show that $\gamma\circ\phi_x\circ\gamma^{-1}\in Inn(G)$. Since $\gamma\in Aut(G), \gamma$ is a homomorphism. Thus For all $g\in G$  $\gamma\circ\phi_x\circ\gamma^{-1}(g)=(\gamma\circ\phi_x)\gamma^{-1}(g)=\gamma(x\gamma^{-1}(g)x^{-1})=\gamma(x)\gamma(\gamma^{-1}(g))\gamma(x^{-1}=\gamma(x)g\gamma(x^{-1}).$  But this is in the group Inn(G). Thus Inn(G) is a normal subgroup of the Aut(G).
Inner automorphisms form a normal subgroup of $\operatorname{Aut}(G)$ For an arbitrary group $(G,\cdot)$ let $\operatorname{Aut}(G) = \{f: G \to G \mid f \text{ is an isomorphism}\}$ be the set of all automorphisms of the group $G$. We assume that $(\operatorname{Aut}(G),\circ)$ where $\circ$ is composition of mappings is a group.  1) Prove that for arbitrary $a \in G$ there is an automorphism $p_a: G \to G;\quad p_a(x) = a^{-1}xa$.  2) Prove, that $\operatorname{Inn}(G) = \{p_a \mid a \in G\}$ is a normal subgroup of $(\operatorname{Aut}(G),\circ)$.
Silencing "Your disk is almost full" notification After upgrading to macOS Sierra, I'm getting the notification that "Your disk is almost full. Save space by optimizing storage.":    The options appear to be to store my files in iCloud, automatically delete files, or manually delete files:    My problem/irritation is that I have 80GB free of my 440GB volume.  The question: is there a way to silence this notification in a (semi-)permanent fashion, or to change the threshold it uses for the notification? What does 'per unit frequency' refer to? We were discussing Kirchoff's theorem on blackbody radiation in class today and the equation was: $$e_{f}= J(f,T)$$  $e_f$ was defined as power per unit area per unit frequency. What does per unit frequency refer to here? I know what per unit time means but don't understand 'per unit frequency'. Anonymous questions and Avatars Determining bare earth DEM from unclassified LAS file? I have data in LAS format with RGB values created from aerial photogrammetry using a UAV. I am  trying to find a solution to extract the bare earth DEM from the point cloud.   I have tried SAGA, Fusion, MCC-LIDAR, but it is seems they need the LAS file to be already classified (which it naturally isn't). Can anyone point me in the right direction with a brief explanation of the process?   Generally, I would need to process about 100 mill points at a time (can tile them if needed). Is there a way to make a fullscreen on VirtualBox? I tried many ways to make a bigger screen in VirtualBox because I do not like the small size of the guest window? Is there is a way to make it big? Can you explain more?  I do not know when we can use them in the sentence.  Can we use them instead of each other?     They have a beautiful home.   They have a beautiful house. When I open nautilus does not open the left bar where it says computer and down markers such as music, documents, videos, downloads. I reinstalled nautilus from the synaptic package manager but nothing happened. What I can do to make that bar visible again? Does one need to be affiliated with a university to publish papers? Assume you have already completed your undergrad and have been working for a number of years. Does one need to be part of a university or a learning institution to publish papers? Term for being unable to see glaring errors after working for some time on a task? What's the de-facto way of reading and writing files in Rust 1.x? The LOTR wiki  Eru Ilúvatar intervened in the history of Arda:   creating Elves and Men removing Aman from the spherical Earth resurrecting Gandalf making Gollum trip      In a letter written by Tolkien, he stated that Eru again intervened, this time in the Third Age, causing Gollum to trip and fall into the fires of Mount Doom while still holding the One Ring, thus destroying it.   Is this correct? If so, what letter is this, and exactly what did Tolkien say? How can I get a brace on the right, not the left, to group cases, using cases? (I know actual cases look better with the brace on the left, but the items I need to group are not literally cases. They are a set of steps in an algorithm which I want to consider as an item.)  The following puts the brace on the left. All I need to do is to flip it to the right.  \begin{align} \begin{cases} a&amp;=bbb\\ c&amp;=ddddddd\\ e&amp;=ffffffffff \end{cases} \end{align}   Thanks for any help with this! Clear Android Market Search History? Any thoughts on how one might clear the Android Marketplace search history? For instance, if I recently searched for an android app in the marketplace, the marketplace search maintains a history list of which apps/keywords that I have recently looked for.    Thanks in advance! Can we get x comment votes per y minutes, rather than one vote per 5 seconds? The bounty I offered on  is now in its grace period and the information box tells me:     you must award this bounty as soon as possible   But there are no answers - what do I do? Can a JavaScript object have a prototype chain, but also be a function? function a () {     return "foo"; }  a.b = function () {     return "bar"; }  function c () { }; c.prototype = a;  var d = new c(); d.b(); // returns "bar" d(); // throws exception, d is not a function   Is there some way for d to be a function, and yet still inherit properties from a? What is the difference between call and apply? I have rigged up a notch filter from . See below: -     Its the first circuit. Now I am trying to calculate its transfer function. I have calculated two equations, but I need the third one:   (Vin-V2)/(1/SC1) = V2/R3 (Vin-V1)/R1 = (V1-Vo)/(R2+(1/SC2))   V1 and V2 are the potentials at the input of the op-Amp. Agent Smith, in the first Matrix movie (The Matrix) talks about how the machines originally created a paradise setting and humans didn't accept it, so it went back and created a more realistic one that humans would accept.  The human bodies, as we've seen, are kept in tanks and the body is kept asleep.  But why did the Matrix need to provide any kind of simulation for the brain to experience?  Why couldn't it have just kept all humans in a medically induced coma?  Or performed something equivalent to a lobotomy on every human?  Why did the Matrix need to spend the resources to keep the humans in good mental shape? So I was in my New Leaf house and I accidentally put on a pair of pants instead of displaying them. Then I went and clicked on myself and I have no option to take off my pants. Ever. I swapped them with another pair for the display, but I can't get those off either. And the weird bit is when I put on the pants nothing was swapped back to my inventory.  How do I take off my pants?