Let $G$ be an abelian group. Show that the mapping $\phi:G\to G$ give by $\phi(x) = x^{-1}$ is an automorphism of $G$. Show that if $G$ were not abelian, then $\phi$ would not be an automorphism.   Okay so this is broken down into two cases;     case $1$: $G$ is abelian then $\phi$ is an automorphism   case $2$: $G$ is non-abelian then $\phi$ is NOT a automorphism   For case $1$: I want to show that $\phi(xy)=\phi(x)\phi(y)$, $\phi$ is $1$-$1$, and $\phi$ is onto, but I'm not sure how to do so.  For case $2$: I want to show that $\phi(xy)\ne\phi(x)\phi(y)$
I have trouble on this question: "Characterize the groups G for which the map $$\iota: G \to G$$ sending $$x \mapsto x^{-1}$$ for all $x \in G$, is an automorphism of $G$." From  post I have seen that $G$ must be abelian. Can anyone propose another characteristic of $G$ that I can derive from the given conditions besides abelian? Thank you so much!!
How do you programmatically set an attribute? Suppose I have a python object x and a string s, how do I set the attribute s on x? So:  &gt;&gt;&gt; x = SomeObject() &gt;&gt;&gt; attr = 'myAttr' &gt;&gt;&gt; # magic goes here &gt;&gt;&gt; x.myAttr 'magic'   What's the magic? The goal of this, incidentally, is to cache calls to x.__getattr__(). Changing my Apple ID's email: address is not available How to show zero before decimal point in bc? echo "scale=3;1/8" | bc   shows .125 on the screen. How to show 0.125 if the output result is less than one? I would like to install Ubuntu alongside Windows 7 on a HP G62 Notebook. Although I have installed  Ubuntu in a dual-boot many times before, I found out that this model has already four partitions.    Partitions:   SYSTEM (NTFS, 199MB, used 66.59MB) Partition without any tag: NTFS (579GM, used 129GB)  RECOVERY (NTFS 16.74 GB, used 2.42GB)  HP_TOOLS (FAT32, 103.34 MB, used 13.23MB)   Since I am not an expert with partitions I would like to get advice on how to do this.  My first idea is this one:   Free some space from /dev/sda2 (I don't know if could also free some other space) delete the HP_Tools partition (I have already created a backup) create an extended partition with the free space in #1 containing three parititions: swap (1gb); / (EXT4, 30GB);  /home (EXT4, 120GB)   Another option is to use wubi instead.  What do you think? Is there any other way to achieve this?  PS: I really think this HP policy of using 4 partitions is not a coincidence  PS: I tried using gparted from the live CD and I got a warning message saying that if I freed some space from /dev/sda2 I could create serious issues in the system I have a Virtual machine using centos 7 with 200GB disk.For some reason my layout is:  df -h Filesystem               Size  Used Avail Use% Mounted on /dev/sda1                100GB  95GB  5GB      Running fdisk -l:  Disk /dev/sda: 204.8 GB, 204803670016 bytes,400007168 sectors  Units = sectors of 1 * 512 = 512 bytes  Sector size(logical/physical): 512 bytes / 512 bytes I/O size(minimum/optimal): 512 bytes / 512 bytes Disk label type: dos Disk identifier: 0x0006c283  Device Boot      Start         End      Blocks   Id  System /dev/sda1            2048   199997439    99997696     Linux   I would like to increase my space on my partision /dev/sda1. Atiyah Macdonald, Exercise 8.3. Let $k$ be a field and $A$ a finitely generated $k$-algebra. Prove that the following are equivalent:   (1) $A$ is Artinian.   (2) $A$ is a finite $k$-algebra.   I have a question in the proof of (1$\Rightarrow$2): By using the structure theorem, we may assume that $(A,m)$ is an Artin local ring. Then $A/m$ is a finite algebraic extension of $k$ by . Since $A$ is Artinian, $m$ is the nilradical of $A$ and thus $m^n=0$ for some $n$. Thus we have a chain $A \supseteq m \supseteq m^2 \supseteq \cdots \supseteq m^n=0$. Since $A$ is Noetherian, $m$ is finitely generated and hence each $m^i/m^{i+1}$ is a finite dimensional $A/m$-vector space, hence a finite dimensional $k$-vector space.  But now how can I deduce that $A$ is a finite dimensional $k$-vector space? What is the smallest positive integer $n$, such that there exist positive integers $a$ and $b$, with $b$ obtained from $a$ by a rearrangement of its digits, so that   $a – b = 11\dots 1$ (The number of '$1$'s equal to $n$)?   Is there any example that can satisfy the equation? Graphing the amount of time a function takes What is the maximum possible length of a .NET string? What is the longest string that can be created in .NET? The docs for the String class are silent on this question as far as I can see, so an authoritative answer might require some knowledge of internals. Would the maximum change on a 64-bit system?  [This is asked more for curiosity than for practical use - I don't intend to create any code that uses gigantic strings!] I'm developing some scripts. One for the  that popped in Ubuntu 16.04 even though pcspkr is blacklisted. Another to fix a  where sound switches off from TV to the laptop.  What is the best directory to put my bash scripts in?  I've searched the Q&amp;A database but I get a bunch of different answers for a bunch of different scenarios.  /usr/local has been recommended by one person because it's in the PATH$. Is that the right thing to do?  Thank you for your advice.    Even though I accepted below answer over a year ago, I ended up using:   /usr/local/bin for scripts needing sudo powers /home/me/bin for scripts running at user level   After creating directory /home/me/bin you need to log out and in again for it to get into your PATH. How many ways can you divide $9$ students into three unlabeled teams where one team contains $4$ people, one contains $3$ people and the last contains $2$ people? Unlabeled, meaning that groups with abc = bca = cba, etc  I understand how to do this if the teams are labeled:  $$\frac{9!}{4!3!2!}$$  But there is a term missing in the denominator when the teams are unlabeled and I'm having difficulty understanding how to calculate how many ways the teams can be organized.  There are $3!$ ways to organize the same first group of $3$, $4!$ ways to organize the same second group of $4$ and $2!$ ways to organize the last group of $2$. Why wouldn't you multiply $3!4!2!$ in the denominator?   For instance:  (ABC, DEFG) = (ABC, DEGF)  = (ABC, DFEG) = (ACB, DFGE), etc How to install Chromium without snap? I just discovered that chromium will only be released as a snap package.  How can I continue to get chromium, without snap? Is this possible without building from source?  Notes:   Please let's keep to the question as stated - not "what do you have against snap?" I'm not asking how to remove snap (I realize that would break Ubuntu)... just how to use chromium as it is now, a deb.     Update  No, this is not a duplicate of that linked question. I don't need to know "why it's a snap". I want to know how to use it without snap.  Right now there is building from source, and some random PPA mentioned below. I hope someone can advise us of trustworthy alternatives. Keyboard key stuck or not being recognized; how to fix This is an attempt to write a canonical QA for this issue, as per the Meta post:   I expect it to be periodically edited with the goal of becoming a comprehensive information resource.  I have a MacBook Pro that has a key(Shift, Return,   Command, P, etc.) that doesn’t respond when pressed or is stuck in the down position.    It's possible that my favorite beverage was spilled (or some other liquid damage).  It's also possible that I have no idea what happened.  It was working yesterday and suddenly today it's no longer working.  Is there any way to bypass or remap the key?  Can I  have the system ignore it completely?  (This question can also apply to external USB keyboards or the Bluetooth keyboards) For a finite set $S$ there are $\vert S\vert!$ orderings of its elements.  What is the cardinality of all orderings of $\mathbb{N}$?  What would $$\vert \mathbb{N}\vert!$$ mean?  Is it $\prod_{n\in\mathbb{N}} \aleph_0=\aleph_0^{\aleph_0}=\aleph_1$?   says the "equivalence classes of well-orderings of the natural numbers" has cardinality $\aleph_1$.  But I can't put together in my mind the connection between well-orderings of $\mathbb{N}$ and all orderings of $\mathbb{N}$.  (Are there non well-ordered orderings of $\mathbb{N}$?)  This leads to wondering what the cardinality of all orderings of $\mathbb{R}$ is.  (Actually my thought process went in the other direction, from $\mathbb{R}$ to $\mathbb{N}$.)  I think knowing the answer to the above question will teach how to determine what $$\vert \mathbb{R}\vert !$$ means, if anything.  I will be happy if you assume the continuum hypothesis to be true for the purposes of this question, but any level of completeness is welcome. How to automate entry of external column data in BCS? I have a document library, Resumes, that has a large number of resumes in it. There is a regular column on that library, called EmployeeID, that is already populated.  I have an external content type ( ECT ) that is mapped to a table that lists EmployeeID and several other columns of information. I added this ECT to the library as an External Column.    I am looking for a good way to automatically go through all items in the document library, and copy the value from the "regular" EmployeeID column into the External EmployeeID column and have it update the other columns from the ECT. Any techniques are useful (PowerShell, Object Model, Workflow?) When we have tried this in the past with workflows the setting of the External Column did not report failure, but didn't actually trigger a lookup of the other data and the value didn't seem to actually get saved.  [Update] I have been working on this and am getting closer. I am using PowerShell to access the list and set the external column ID field to the value I want it to be.  The problem is that it doesn't seem to be triggering a lookup of the external data when that field is set.  my "list" (a document library) has a few normal fields and a set of external columns mapped to an ECT "EmplData".  When you use the GUI to add or edit the properties of an item, it works. When I use PowerShell to do the following:  $spList = $spWeb.Lists["Resumes"] $spListItem = $spList.GetItemById(1) $spListItem["EmplData_ID"] = "00300"    $spListItem.Update()   Sometime the external ID field gets updated when I the refresh the web page, other times it says "(blank)". In all cases it does not load in the other fields.  Is there some Update method that updates the rest of the elements?  I gather from research that these are called "secondary fields", is the method to update the secondary fields based on the external item ID available in PowerShell cmdlets someplace?  [Update] I am finding that if I use:  $spListItem["EmplData"] = "000300"   instead of $spListItem["EmplData_ID"] = "00300" it gets around the problem of the column showing "(blank)".   I still need to figure how to make it update the other external columns (also known as "secondary columns" I think) Gandalf (as with all the Istari) did not have a corporeal form before coming to Middle-earth and was essentially ageless. We also know that Gandalf did age, albeit very slowly, while on Middle Earth. By the end of the Third Age (before battling Durin's Bane), he had the appearance of an old human male (non-Númenórean, etc) in his 70's or 80's. This leads me to wonder:  Is there any indication in Tolkien's writings that describes Gandalf's appearance (in human-looking years) when he arrived in Middle Earth? In other words, was Gandalf the Grey originally "Gandalf the Tall Dark &amp; Handsome in his 20's"?  I'm not talking about his actual age, but rather his appearance in human years. I have just started studying implication in mathematics. So, you'll probably have an idea of where my confusion lies right from the get go.  In the truth table, where $A \implies B$ we obtain this result:  A   B   T/F ___________ T | T | T T | F | F F | T | T F | F | T   Now, my confusion here is regarding why $A\implies B$ given A is false, and B true. The others I understand. The first and last one are obvious, the second one implies, to me anyway, that given A implies B, the truth of B rests upon the truth of A, B is false, A is True, which cannot be, thus not B given A is false.  Now... then, why is B true despite the fact that A is false? Or rather, why is the statement B given not A, True. The truth of B is implied from the truth of A, but A is a false statement. So it's very contradictory that this may be, and in fact, if you go through a real life example, you would find that the third statement seems to be false.  Say $A$ is the statement that it is dark outside, and $B$ the statement that the sun is not on our side of the earth. If it is not dark outside, than stating that the sun is on the other side of the planet must be wrong (in my example i'm going to claim that the sun is the only light source).  Side note:  I just had a thought, if A is false, does that mean you can conclude... mmm, whatever? I mean, B is implied via the truth of A, if A is false, B is NOT implied, but then B could be EITHER true or false, I suppose. Is this correct?  However, would that not mean that A does not imply anything when false. Would that not set the 3rd and 4th statement to be undefined? Where are my installed applications? Sometimes, a question is perfectly solved by suggestion or actual resolution given in "comments" under it. In such case, how to mark that answer (comment) as "Accepted Answer"? Due to this, the question remains unanswered even if solution is provided in comment. How do I give players already enchanted Tools/Armour/Weapons I'm making a map where I want players to be able to choose there kits/classes. The problem I'm having with it though is giving players already enchanted Tools/Armour/Weapons.  I'm not sure if you can use the /give [player] [item] [amount] because you'll have to use the whole {} thing and I'm not sure how to make enchanted Tools/Armour/Weapons using that.  Can anyone help?