Let $K$ be a field, $G$ a finite group and $V$ a finite-dimensional $KG$-module. Consider the endomorphisms of $V$, i.e. linear mappings $\phi : V \rightarrow V$ such that $g\phi(v) = \phi(gv)$ for all $g \in G, v \in V$. Now, if the characteristic $\operatorname{char} K$ is coprime to $|G|$ (or $0$), then by Maschke's theorem, we can decompose $V$ into a direct sum of irreducible $KG$-Modules, and easily show the following property: if $V$ isn't irreducible (i.e. it is a direct sum of at least 2 summands), it has an endomorphism that is not simply a scalar multiple of $\operatorname{id}_V$: we can permute the summands, which is clearly an endomorphism (even an automorphism!) of $V$. Equivalently, if the only endomorphisms of $V$ are scalar multiples of $\operatorname{id}_V$, then $V$ is irreducible. My question is now, whether this property holds in the case that $\operatorname{char} K$ and $|G|$ share some divisor. Since Maschke can fail in this case, my intuition would be that this shouldn't hold, but I cannot seem to construct a counterexample. For an &quot;easy&quot; example, I thought about using $V = \mathbb{F}_2^2$ over $K = \mathbb{F}_2$. To ensure that the only endomorphisms of $V$ are scalar multiples of $\operatorname{id}$, $G$ should at least contains elements that act on $V$ like the matrices $\begin{pmatrix} 0 &amp; 1 \\ 1 &amp; 0 \end{pmatrix}$ and  $\begin{pmatrix} 1 &amp; 1 \\ 0 &amp; 1 \end{pmatrix}$, since the only matrices that commute with both of these are the identity and the zero-matrix. However, this also has the consequence that $V$ has no non-zero $KG$-subspaces. If the property does hold in this case as well, how would I go about proving this?
Let $G$ be a finite group and $F$ a field with $\mathrm{char}(F)=0$ or coprime to $|G|$. Let $V$ be a $FG$-module in a way that every $ FG$-homomorphism $ f : V \to V $ is given by $f(x)= \lambda x $. Then $V$ is irreducible.   I already managed to proof this by contradiction, using Maschke's Theorem. We can  write $V$ as $U\oplus W$ and get a $FG$-homomorphism $\pi:U\oplus W \to U\oplus W$, $\pi (u+w)=u$, then $\pi(u+w)=\lambda (u+w)=u $ then either $U$ or $W$ are trivial and thus $V$ is irreducible.  My question is if this statements still holds if the characterictic of $F$ divides the order of $G$, since I only used this fact in my proof to use Maschke's theorem.
Counting the number of propagating degrees of freedom in Lorenz Gauge Electrodynamics Is there a way to get the path for the assembly in which the current code resides?  I do not want the path of the calling assembly, just the one containing the code.   Basically my unit test needs to read some xml test files which are located relative to the  dll.  I want the path to always resolve correctly regardless of whether the testing dll is run from TestDriven.NET, the MbUnit GUI or something else.  Edit: People seem to be misunderstanding what I'm asking.  My test library is located in say      C:\projects\myapplication\daotests\bin\Debug\daotests.dll   and I would like to get this path:     C:\projects\myapplication\daotests\bin\Debug\   The three suggestions so far fail me when I run from the MbUnit Gui:   Environment.CurrentDirectory gives c:\Program Files\MbUnit System.Reflection.Assembly.GetAssembly(typeof(DaoTests)).Location gives C:\Documents and Settings\george\Local Settings\Temp\ ....\DaoTests.dll System.Reflection.Assembly.GetExecutingAssembly().Location gives the same as the previous. What’s the correct URL for placing a call on an iPhone? Allow embedded HTML5 YouTube video How to use exclamation marks in Minecraft commands? I know that you can use exclamation marks in Minecraft commands to check for an entity that does not have a specific tag. For example, you can use /kill @e[type=!Player] to kill all entities except for the player. I have, however, encountered two problems with using exclamation marks that I can't find a solution for.  For my first problem, I've set up a series of command blocks to summon shootable fireballs above the player’s head when a certain item is being held;  /scoreboard players add @a Fireball 1 {SelectedItem:{tag:{display:{Name:"Summon Fireball"}}}} /execute @a[score_Fireball_min=1,score_Fireball=1] ~ ~ ~ /summon Fireball ~ ~2.5 ~ {direction:[0.0,0.0,0.0]} /scoreboard players set @a Fireball 0   I used a comparator and a redstone torch so that the last command would activate when the first command would deactivate, that is, when the player is not holding the item.  The problem is, that when I move far away from the setup, the redstone stops working, leaving me unable to summon any new fireballs. Instead, I'm trying to use a command block to check when anything except for item is being held;  /scoreboard players set @a Fireball 0 {SelectedItem:{tag:{display:{Name:!"Summon Fireball"}}}}   I’ve either put it at the wrong place, or you simply can’t use exclamation marks in NBT tags. Could someone tell me where I should put the exclamation mark, or in the latter case, how I can check for anything except for a specific item?  Further, I’ve tried using exclamation marks to set the score of all entities, including the ones with a blank score, but except for the ones with a score of 2 or higher, to 0;  /scoreboard players set @e[score_arrows_min!=2] Arrows 0 /scoreboard players set @a Arrows 1 {SelectedItem:{tag:{display:{Name:"Rain of Arrows"}}}} /scoreboard players set @e[type=Arrow] Arrows 2 {inGround:1b}   The code does change the scores of entities without scores to 0, but it doesn’t exclude the arrows with a score of 2. Could anyone tell me what I’m doing wrong here? I need a formula for conditional formatting to make an Excel 2010 spreadsheet alternate highlighting when a single column changes.  This question, "", is exactly what I want to do, but the suggested solution didn't work for me--it just highlights the first row of each matching value.  I know how to use conditional formatting, but for the life of me I can't figure out or find any pointers on the net so far to make this work.  Ideas?  Here is a link to a picture of how I want my spreadsheet to look when I'm done.  Basically I want every other Disp Number value row to be highlighted.  Seems like it would be a common thing to want to do, and I've seen this asked for in various places, but people struggle with making it work. Convert date to another timezone in JavaScript Remove vote UI elements beside own question and answer posts Currently, the system makes you think you can upvote/downvote your own question/answer but then you get notified that you can't. Why have the up/down vote buttons there then?   In general, if some operation is not allowed, then don't put UI for that up or make it disabled so it is apparently disabled - no post operation error panels please?  ps: hope this is not a dupe, but if so, just comment it and I will remove  (+a joke: a user notification panel message: I deleted this file. Is this a problem? Yes/No.) How can I check if the editor that is currently being used is Gutenberg in a WordPress plugin?  I need this because Gutenberg lacks post_submitbox_misc_actions, so I need a fallback which will only be used if the current editor is Gutenberg. Can you enter the Schengen area with a UK visa? Prove or the disprove the existence of a limit of integrals Let $A \subset \mathbb{R}^2$ be the annulus defined, in polar coordinates, as  $$ A = \{ (r, \theta) \in [0, \infty) \times [0, 2\pi) : 2 \leq r \leq 3 \} \, .$$ Let $F: A \to A$, $F:(r, \theta) \mapsto (r, r \theta \text{ (mod } 2\pi))$. Let $g : A \to \mathbb{R}$ be a continuous function. Discuss the existence of the limit $$ \lim_{n \to \infty} \int_A g \circ F^{(n)}(x) \, \text{d}x \, ,$$ where $F^{(n)} = F \circ F \circ \dots \circ F$ denotes the composition of $F$ with itself $n$ times.  I cannot find a way to attack this problem. Here's what I tried: if $g \circ F^{(n)}$ converged, I think we could take the limit under the integral sign. So I tried to show the convergence of $F^{(n)}$. If it converged, then the limit function must send every point to a fixed point of $F$. So I determined the fixed points of $F$. But I don't think $F^{(n)}$ converges. What is the best (idiomatic) way to check the type of a Python variable? I need to know if a variable in Python is a string or a dict.  Is there anything wrong with the following code?  if type(x) == type(str()):     do_something_with_a_string(x) elif type(x) == type(dict()):     do_somethting_with_a_dict(x) else:     raise ValueError   Update: I accepted avisser's answer (though I will change my mind if someone explains why isinstance is preferred over type(x) is).  But thanks to nakedfanatic for reminding me that it's often cleaner to use a dict (as a case statement) than an if/elif/else series.  Let me elaborate on my use case.  If a variable is a string, I need to put it in a list.  If it's a dict, I need a list of the unique values.  Here's what I came up with:  def value_list(x):     cases = {str: lambda t: [t],              dict: lambda t: list(set(t.values()))}     try:         return cases[type(x)](x)     except KeyError:         return None   If isinstance is preferred, how would you write this value_list() function? Old Sci-Fi book with skin crystalizing disease and characters with psi powers Help me identify this fantasy book about two near-omnipotent wizard run empires Summary table with a field that shows components of each summarized record I currently have the Geometry Dash lite version and I want to buy the full version. Can I login with my account of Geometry Dash lite and keep everything I already had? It would suck if I had to redo everything. How to understand the LaTeX warning “there were undefined references”? No, this is not a , nor is it a solicitation on  to backup or  and only a little bit about Where to backup and When to do it.  This is basically a strategy question: what, where and when all together. "How" is not really relevant: there are tons of back-up programs out there and they all do the same basic thing: make copies of data.  But the real question is: are these back-up programs making copies of the data important to you?  How should you install&amp;run Ubuntu while safeguarding your data?  There is no "one size fits all" when it comes to computers, Ubuntu versions or a backup strategy, therefore I've split this up into 5 basic user types with each their own answer:       (or you're running in UEFI mode!)   And as the above does not fit everyone, one for the impatient:   !    So click one of the above links and go to your user type!  P.S. If you're running a server farm and you've read this question so far, I'm sorry: this is about a single PC, not a network, nor a server farm. C# split string but keep split chars / separators I'm splitting a string by three different characters but I want the output to include the characters I split by. Is there any easy way to do this? What does really attracts a water stream to a charged object? I saw  by  and I was not 100% sure about the third phenomena, a statically charged object attracting a stream of water, especially because he explicitly mentioned that it is not due to the  of the water. A week later he uploaded an  all five phenomena and he says it is because of repelled equally charged ions, back up the water flow, such that the water actually falling along the charged object would be oppositely charged and therefore attracted. Here is a still image from the video illustrating this:    However this explanation seems strange to me, since this would mean that this way you could build up a charge in the water which has exited the tap (as well as in the water inside the tap as well). Therefore it should get harder to push those ions back over time, and therefore there also would be a decrease in the attraction of the water (assuming there is no other way to discharge it). I have tried a simple similar experiment, where I catched the water leaving the tap in an insulated bowl, but I did not notice a decrease in attraction.  There a couple of reasons why my experiment does not rule out his theory:   I was holding the charged (plastic) object by hand, which caused that its distance to the water stream varied, making it difficult to observe a decrease in attraction. I only conducted my experiment for roughly 30 seconds, which might be enough time for a opposite charge to build up.   The reason that he mentioned why permittivity can't cause this phenomena is because the electric field would be homogeneous and therefore the attractive and repulsive force of both dipoles of a water molecule would cancel out. However the electric field would only be homogeneous if the charged object would be an infinitely large plate, which it isn't. Thus water molecules would be attracted to the charged object. In order to make the permittivity theory more plausible I thought of maybe doing a order of magnitude calculation to see if it could generate a large enough force would be able the water stream by the observed amount. However I do not know what a good guess would be for variables such as the amount charge on the charged object and how the electric field would fall off over distance.   So, my question is whether his theory is really correct, or if the permittivity of the water can cause it.