$G$ is a group with $e$ as it's identity element. The commutator group $[G,G] = &lt;[x,y]|x,y \in G&gt;$ such that $[x,y]=x^{-1}y^{-1}xy \: \: \: \forall x,y \in G$  Show that $[G,G]\triangleleft G$  Solution: If $[G,G]\triangleleft G$, then $g^{-1}[x,y]g$ must belong to [G,G].  $$g^{-1}[x,y]g$$ $$g^{-1}x^{-1}y^{-1}xyg$$  I am stuck here. I am not sure how to proceed since we have three elements...  Thanks for the help!
It is suggested as an exercise in Serge Lang's book "Algebra" to show that the commutator subgroup $G^c$ of a group $G$ is a normal subgroup.   I'd like to do that but I am afraid I need help,   I think the first thing I need to figure out is how a general element in the commutator subgroup looks like, so that I can check that the defining condition for normality is satisfied.  That is, supposing for a moment that a general element in $G^c$ is denoted by $g$, I need to show that $aga^{-1} \in G^c,$ for all $a \in G$.  But here I get stuck, first because I am unsure how to write a general element in $G^c$ - a simple product in $G^c$ is of the form $xyx^{-1}y^{-1}aba^{-1}b^{-1}$ where $a,b \in G$. I cannot see a way to simplify this - I am sure there is one, but somehow I am blind today.  The second thing then is, even if one tries out the conjugation of a simple element like $xyx^{-1}y^{-1}$ in $G^c$, again not simplification offers itself easily I think .. what am I missing ?  An alternative would be to find a homomorphism of $G$ whose kernel is precisely $G^c$ - here I tried to think of this as a map $G \times G \to G$ but whatever I cook up is not a homomorphism.  Thanks for your hints !!
Why are There No "Triernions" (3-dimensional analogue of complex numbers / quaternions)? Since there are  complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions").   Yet no one uses these. Why is this? Proof of when is $A=X^TX$ invertible? Say we have an $n\times m$ matrix $X$. What are the specific properties that $X$ must have so that $A=X^TX$ invertible?  I know that when the rows and columns are independent, then matrix $A$ (which is square) would be invertible and would have a non-zero determinant. However, what confuses me is, what sort of conditions must we have on each row of $X$ such that $A$ would be invertible.   It would be very nice to have a solution of the form:   when $n &gt; m$ then $X$ must have... when $n &lt; m$ then $X$ must have... when $n = m$ then $X$ must have...   I think in the 3rd case we just need $X$ to be invertible but I was unsure of the other two cases. String representation of an Enum I have the following enumeration:  public enum AuthenticationMethod {     FORMS = 1,     WINDOWSAUTHENTICATION = 2,     SINGLESIGNON = 3 }   The problem however is that I need the word "FORMS" when I ask for AuthenticationMethod.FORMS and not the id 1.  I have found the following solution for this problem ():  First I need to create a custom attribute called "StringValue":  public class StringValue : System.Attribute {     private readonly string _value;      public StringValue(string value)     {         _value = value;     }      public string Value     {         get { return _value; }     }  }   Then I can add this attribute to my enumerator:  public enum AuthenticationMethod {     [StringValue("FORMS")]     FORMS = 1,     [StringValue("WINDOWS")]     WINDOWSAUTHENTICATION = 2,     [StringValue("SSO")]     SINGLESIGNON = 3 }   And of course I need something to retrieve that StringValue:  public static class StringEnum {     public static string GetStringValue(Enum value)     {         string output = null;         Type type = value.GetType();          //Check first in our cached results...          //Look for our 'StringValueAttribute'           //in the field's custom attributes          FieldInfo fi = type.GetField(value.ToString());         StringValue[] attrs =            fi.GetCustomAttributes(typeof(StringValue),                                    false) as StringValue[];         if (attrs.Length &gt; 0)         {             output = attrs[0].Value;         }          return output;     } }   Good now I've got the tools to get a string value for an enumerator. I can then use it like this:  string valueOfAuthenticationMethod = StringEnum.GetStringValue(AuthenticationMethod.FORMS);   Okay now all of these work like a charm but I find it a whole lot of work. I was wondering if there is a better solution for this.  I also tried something with a dictionary and static properties but that wasn't better either. Setting up PyCharm to reference ArcPy? How would I setup  to recognize import arcpy?   I'd like to use PyCharm to develop and debug geoprocessing tool. Building a graph from OSM? I spent days looking in all questions here and trying tutorials. I need to build a graph for a region (It will be my input in simulation) I downloaded Postgresql and Postgis and osm2pgsql. Now i can load .osm files into a database. I want to have a complete database with roads ids so i can generate a table of edges and nodes then build my graph  nodes = intersections and edges = roads for a city using python.  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Opposite of Alternatives Since drush 9 (drupal 8.4 requirement), alias @sites does not seem to be implemented anymore (or yet ?).  $ drush @sites cr The alias @sites could not be found.   Is this a way to get the same behavior that does not consist of typing my 300 alias in a config file ? Accessing private member variables from prototype-defined functions Is there any way to make “private” variables (those defined in the constructor), available to prototype-defined methods?  TestClass = function(){     var privateField = "hello";     this.nonProtoHello = function(){alert(privateField)}; }; TestClass.prototype.prototypeHello = function(){alert(privateField)};   This works:  t.nonProtoHello()   But this doesn’t:  t.prototypeHello()   I’m used to defining my methods inside the constructor, but am moving away from that for a couple reasons. Pre-declare all private/local variables? The &quot;Your Spellbook&quot; sidebar for the wizard includes the following line:  Copying a Spell into the Book. When you find a wizard spell of 1st level or higher, you can add it to your spellbook if it is of a level for which you have spell slots and if you can spare the time to decipher and copy it.  Does this mean that a Wizard 1/Cleric 19 can copy any wizard spell they find into their spellbook? Is there anything preventing them from casting these spells? Is lemonade better when made with simple syrup? On a hot summer day and with a basketful of lemons I often want a nice cold glass of lemonade. A lot of recipes suggest making lemonade with simple syrup, which requires cooking, whereas other recipes just have stirred-in sugar.  What exactly are the benefits of making lemonade (or limeade) with simple syrup versus just stirring in sugar? How to raise a complex number to the power of another complex number? How do I calculate the outcome of taking one complex number to the power of another, ie  $\displaystyle {(a + bi)}^{(c + di)}$? In the Linux system, we can use bellow command to enable packets forwarding:  ~ # sysctl net.ipv4.ip_forward=1 net.ipv4.ip_forward = 1   but how can I realize the same function in Windows?  I use Windows Server 2008 R2. Why didn't anybody Summon the Triwizard cup in the maze? We know that the Triwizard cup itself was Summonable. After all, Harry did Summon it in the graveyard.  So why didn't any of the contestants Summon it while in the maze?  Canonical answers, please. $f$ strictly increasing does not imply $f'>0$ We know that a function $f: [a,b] \to \mathbb{R}$ continuous on $[a,b]$ and differentiable on $(a,b)$, and if $f'&gt;0 \mbox{ on} (a,b)$ , f is strictly increasing on $[a,b]$. Is there any counterexample that shows the converse fails?   I have been trying to come up with simple examples but they all involve functions that are discontinuous or has derivative $f'=0$ which does not agree with the hypothesis hmmm Is there a catch-all place on the Stack Exchange network where one can post any question that does not obviously belong to the Stack Exchange sites? Is there a catch-all place somewhere on the Stack Exchange sites where one can post a (any) question that does not obviously belong to any of the Stack Exchange sites?  Sometimes I have questions but I don't want to ask them on other sites (no need for names). I prefer to post it on the Stack Exchange network for various reasons.  If there isn't, are there plans for one? I imagine that if such a site exists, questions on that site can also be moved to other more specific sites when they exists, and it's not to be used to post questions which obviously belong to a specific Stack Exchange site. I have a netbook that I want to install Ubuntu onto. I do have a good Mac computer that I can make a bootable USB drive from, but I'm confused on how to go about making it.