Let $H$ be a subgroup of $G$.   $S=\{g\in G\mid ghg^{-1}\in H,\forall h\in H\}$.   I know that $N_G(H)\subset S$. But I can't seem to find any counterexamples for the other inclusion. Any ideas?
Let $G$ be a group, $H\subseteq G$ a subgroup and $a\in G$ an element of the group. Is it possible that $aHa^{-1} \subset H$, but $aHa^{-1} \neq H$?   If $H$ has finite index or finite order, this is not possible.
Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$ Suppose $W$ is a subspace of the finite-dimensional vector space $V$ and $\dim W = \dim V=n$  Let $\{\overrightarrow w_1,\ldots,\overrightarrow w_n\}$ be a basis for $W$ and $\{\overrightarrow v_1,\ldots,\overrightarrow v_n\}$ be a basis for $V$.  The set $\{\overrightarrow w_1,\ldots,\overrightarrow w_n,\overrightarrow v_1,\ldots,\overrightarrow v_n\}$ still spans $W$. Since $W$ is a subspace of $V$, $W\subset V$ and all the $w_i$'s can be written in terms of $\{ \overrightarrow v_{1},\ldots,\overrightarrow v_n\}$.  $\therefore$ We can remove one by one all the $w_i$'s from the set $\{\overrightarrow w_{1},\ldots,\overrightarrow w_n,\overrightarrow v_1,\ldots,\overrightarrow v_n\}$ while still having it span $W$, which leaves us with the set $\{\overrightarrow v_1,\ldots,\overrightarrow v_n\}$. We know it to be linearly independent since it is a basis for $V$. Therefore the basis of $V$ is also a basis of $W$. From there on, it is trivial to prove double inclusion $W\subset V$ and $V\subset W$ which gives us $W=V$.  Is my approach correct? Unique (non-repeating) random numbers in O(1)? What are some alternatives to "hover" on touch-based devices? Came up recently in a 37signals podcast:    The straightforward, "brute-force" solution is to just use the first touch on a UI object as the hover.    It works. But are there other, more elegant solutions? For instance, could the hardware ever be made to support actual finger hovering? A projection operator $P$ is defined as $P^2$=$P$. Use this definition to find the eigenvalues of this operator.   In this question is it necessary to define what the projection operator is? And won't the eigenvalue just be zero? How do I escape a percentage sign in T-SQL? Is there a shortcut for switching between these three modes? How to get scroll position from GridView? probability that exactly two accidents happen in any day of a week if 7 accidents happen in a week If 7 accidents happen in a week, what is the probability that exactly 2 accidents happen in any day of the week.  I tried to work it this way.  2 accidents out of 7 can be taken in ${7}\choose{2}$$= 21$.  Now this can happen in any of the 7 days, so we have $7 \times 21$.  Now, the remaining 5 can happen in $6^5$ ways.  So the probability is  $\frac{7\times 21 \times 6^5}{7^7} = \frac{3 \times 6^5}{7^5}$   But this gives me a value greater than 1.   I know I am doing something wrong.  Can anyone help me ? The mystery of the tagged answer - and more It is (currently)   to tag answers. Yet, if I query , then  does have a tag, namely javascript. I'm at a loss: how did that happen?&dagger;  SELECT Id, Tag FROM Posts, PostsTags  WHERE Posts.PostTypeId = 2 AND Posts.Id = PostsTags.PostId   Edit: The are also five answers (, , ,  and ) with a view count of 1 instead of 0:  SELECT Id, ViewCount FROM Posts WHERE PostTypeId = 2 AND ViewCount &lt;&gt; 0   These results also hold through at least January 10.    &nbsp; &dagger;: PostTypeId = 2 means that the post is an answer, not a question (PostTypeId = 1). Algebraic proof of $\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$ I can't figure out an algebraic proof for the following identity:  $$\sum_{i=0}^k{{n \choose i}{m \choose {k-i}}}= {{m+n}\choose k}$$  Combinatorical solution:  We can see that as choosing some from $n$ and the rest of $k$ from $m$, thus $k$ in total.  Or we could just choose $k$ from the union. Is a DNA molecule a single strand of polynucleotide or two of them linked together? jQuery - Is it okay to use $('#ElementId') everytime? Runtime error : ERROR 000358: Invalid expression using SelectLayerByAttribute in ArcPy? I cannot for the life of me persuade SelectLayerByAttribute to work (I need to automate this for a complex script).  Working in ArcMap's python console on already loaded shapefile, the format below:  import arcpy from arcpy import env arcpy.env.workspace = r"C:/Users/.../working" arcpy.SelectLayerByAttribute_management("layer", "NEW_SELECTION", " 'ATTRIBUTE' = 'Criterion' ")   results in error:     Runtime error : ERROR 000358:   Invalid expression Failed to execute (SelectLayerByAttribute).   An alternative method:  import arcgisscripting import arcpy gp = arcgisscripting.create(9.3) gp.workspace = "C://...//working" sFile = "C://...//working//shapefile.shp" lyr = "layer" gp.makefeatureLayer_management(sFile, lyr) gp.toolbox = "management" gp.SelectLayerByAttribute("layer", "NEW_SELECTION", " 'ATTRIBUTE' = 'Criterion' ")   .. selects everything, including features that don't meet the criterion!  Does anyone have any idea what's going wrong?  I've read some pages on , but found no solution that works here. The working directory is correct. The attribute field and criterion are correctly specified (case sensitive etc), etc. Its very confusing. Consider the functional equation problem $$ f : \mathbb R \to \mathbb R \text , $$ $$ f \left( a ^ b \right) = f ( a) ^ { f ( b ) } \text , $$ when $ a , b \in \mathbb R $, $ a , b \ge 0 $. So far the only solution I have is the trivial $$ f ( x ) = x \text . $$ Does there exist any other possible solution? Even if the $ f $ doesn't have a closed form in terms of elementary functions, is there some way I could derive a series or an alternative expression for it? I tried to get a little more creative: $$ f ( w ) = f \left( w ^ { \frac 1 w } \right) ^ { f ( w ) } \text , $$ which nests deep into $$ f ( w ) = f \left( w ^ { \frac 1 w } \right) ^ { f \left( w ^ { \frac 1 w } \right) ^ { f \left( w ^ { \frac 1 w } \right) ^ \vdots } } \text , $$ which is equivalent to $$ \ln \big( f ( w ) \big) = \ln \bigg( f \left( w ^ { \frac 1 w } \right) \bigg) f ( w ) \text . $$ I was hoping to generate some sort of series using this. Suppose we have an object in space. say gravity is negligible, and the thing has a rocket on both sides. the thing fires one rocket for, say, 5 seconds. now it's moving. The object has kinetic energy. Now the other rocket fires for 5 seconds. Then it's stationary. where does the energy go? I know that some of the energy from the rockets would just go into space as heat and kinetic energy of the hydrogen particles flying out, but how does there seem to be more energy in the situation if the first rocket just goes for 10 seconds than if one goes for 5 and the other goes for 5? Why should users have to press Return twice to insert new lines? When I post a question and want to add a new line, I have to press Return two times, or else the next text will appear on the same line.  Is this really supposed to behave this way? If so, why? There is a kind of situation that seems to occur somewhat commonly, that goes something like:   Figure out some way to handle a particular situation do it regularly, and make a habit of it, as it seems to work OK … … and only much later actually realize that it very much isn’t OK, that it’s been doing harm all the time, and that it’s too late to fix anything.   Does there exist any good words or phrases for this kind of situation? How can I make sure that a certain OLEDB driver is installed when I start my application? I use ADO from Delphi and would like to display a descriptive error message if the driver is missing. The error that's returned from ADO isn't always that user-friendly.  There are probably a nice little function that returns all installed drivers but I haven't found it. Which of these two phrases has more negative undertone?      "The interview I had was not very good"    or      "The interview I had was not good" Can I mount the root (system) filesystem as writable in Big Sur?