I want to show that     If $G$ is a finite $p$-group, then the number of nonnormal subgroups of $G$ is a multiple of $p$.   I think I need to consider a conjugation action. But then?
Let $G$ pe a $p$ group. I have to show that         the number of nonnormal subgroups is divisible by $p$       the number of subgroups differs from the number of normal subgroups by a power of $p$.   Are there any theorem that can help me to prove this ? We have discussed the Sylow theorems but I don't know how to apply them - if those are the theorems I need. (Does this theorem also hold for infinite groups $G$ ?)
SQL Server Max Memory not limiting use of RAM I would like your input on this. I have a sql server 2008r2 Ent. Ed. 64bit with 16 cores and 64GB RAM. There is one instance of SQL server patched fully as of 20111014.  The max ram is set to 60000MB. Amount of free ram is 0 according to task manager after a few days online.  If I change the max ram to below 53GB it will after a few days to stabilize and have some free ram.   It is the sql process that does allocate the ram according to task manager. How do i come to terms with what the problem really is? It goes without saying that i did alot of testing already but havn't solved this to my liking yet. and ohh we do not get the typical memory starvation lagging when the available ram is down to 0 free.  Update 1:  Inspired by another Q/A related to RAM on this page  . I used these two to see what the RAM is being used for.  SELECT TOP ( 10 )         [type] AS [Memory Clerk Type] ,         SUM(single_pages_kb) AS [SPA Mem, Kb] FROM    sys.dm_os_memory_clerks GROUP BY [type] ORDER BY SUM(single_pages_kb) DESC OPTION  ( RECOMPILE ) ;  SELECT  DB_NAME(database_id) AS [Database Name] ,         COUNT(*) * 8 / 1024.0 AS [Cached Size (MB)] FROM    sys.dm_os_buffer_descriptors --WHERE   database_id &gt; 4 -- system databases --        AND database_id &lt;&gt; 32767 -- ResourceDB GROUP BY DB_NAME(database_id) ORDER BY [Cached Size (MB)] DESC OPTION  ( RECOMPILE ) ;   The amount used shown by these are  first select 7948432    Kb second one 44030,57812  MB that is a total of about 52GB used by sql server... so where did the rest of my RAM go? :-) Task manager show right now cached 363, available 401, free 40 and sqlservr.exe has Memory private set 64 459 656. Max Ram set to 60000MB as before. How to run a specific program as root without a password prompt? I need to run something as sudo without a password, so I used visudo and added this to my sudoers file:  MYUSERNAME ALL = NOPASSWD: /path/to/my/program   Then I tried it out:  $ sudo /path/to/my/program [sudo] password for MYUSERNAME:    Why does it ask for a password? How can I run/use commands as root with a non-root user, without asking for a password? Zero module of differentials implies finite extension? Is it possible to insert arguments into functions when they're used like Function@ or //Function? I'm trying to debug a Python script by placing a print("foo=" + foo) statement in my code.  But where does the output appear?     It doesn't appear on the Python Console, and it doesn't appear on the area that appears when you drag the lower edge of the infobar downwards (what is the name for this area? My script errors show up in red here.) Weighting features prior to SVM I'm building an object detector using HOG features and linear SVM. Some of the regions of the object are more "distinctive" so I would like to give more weight to the features extracted from those regions. (e.g. imagine we want to detect between mugs and glasses, then the most "distinctive" part is the mug's handle)  How could I do this? For sure there should be a more intelligent way than just replicating the features of those "distinctive" regions. Is there any evidence or reason why FIFA is so against adding instant replay to soccer? I find myself seeing many unlucky or just plain wrong decisions made by refs that could be corrected with instant replay. Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$), are true, why aren't logarithms defined for negative numbers?  Why can't $b$ be negative? Take $(-2)^3 = -8$ for example. Turning that into a logarithm, we get $3 = \log_{(-2)}{(-8)}$ which is an invalid equation as $b$ and $x$ are not positive! Why is this? Divergence of $\sum_{n=1}^{\infty} \frac{1}{2+i^n}$? I notice that,  $i^n = i, \quad if \text{ } n \equiv \text{ } 1 \text{ } mod \text{ } 4$  $i^n = -1, \quad if \text{ }n \equiv \text{ } 2 \text{ } mod \text{ } 4$  $i^n = -i, \quad if \text{ }n \equiv \text{ } 3 \text{ } mod \text{ } 4$  $i^n = 1, \quad if \text{ } n \equiv \text{ } 0 \text{ } mod \text{ } 4$  so the series $\sum_{n=1}^{\infty} \frac{1}{2+i^n}$ seems to be an alternating series that does not converge to any point in the complex plane. Is there a rigorous way to prove this divergence? How would you write a proof to this divergence or convergence (if I am wrong)?  Thanks! Summation of series $\sum_{n=1}^\infty \frac{n^a}{b^n}$? Projection of a point onto a simple convex polytope I have a convex polytope described by the intersection of a hypercube and two parallel lower and upper hyperplanes cutting through the hypercube. I want to find the projection of a point onto the polytope. I read the question  and see that the general problem of finding a projection onto a convex polytope can be solved with quadratic programming. However, taken alone, projection onto a hypercube or a hyperplane is trivial. For the hypercube, each dimension is defined by a lower+upper bounds and we just pull each dimension within those bounds, which take $O(n)$ time in $n$ dimensions. For the hyperplane just find the projection onto it. So I'm wondering whether there is simpler method for finding such a projection, maybe somehow combining the two independent projections. "I let him do it" and "I allow him to do it", why exactly does one require 'to'? In the C and C++ programming languages, what is the difference between using angle brackets and using quotes in an include statement, as follows?   #include &lt;filename&gt;  #include "filename" Distance function is continuous in topology induced by the metric The question is (from Topology without tears) that: Let $(X,d)$ be a metric space and $\tau$ the corresponding topology on $X$. Fix $a \in X$. Prove that the map $f:(X,\tau) \rightarrow \mathbb{R}$ defined by $f(x) = d(a,x)$ is continuous.  My first attempt was that let an open ball, $B_{\epsilon}(a)\subset \mathbb{R}$ and show that $f^{-1}(B_{\epsilon}(a)) \subset \tau$. Since $f(x) = d(a,x)$, $f^{-1}(B_{\epsilon}(a))=B_{\epsilon}(a) \in \tau$.   I am pretty sure the last sentence is wrong. I am not sure if every open ball is an open set in topology induced by metric space.  Please give me some hint and direction. Thanks In an HTML table, the cellpadding and cellspacing can be set like this:  &lt;table cellspacing="1" cellpadding="1"&gt;   How can the same be accomplished using CSS? Why are these numbers not equal? Retrieving HTTP status code from loaded iframe with Javascript Is it possible to have connecting loop lines (like algorithm2e) in algorithmic? I am using the algorithmic package to typeset an algorithm. I would like to have connecting lines between the beginning and end of a loop or conditional like in the algorithm2e package. Is there an easy way to accomplish this, or am I better off switching to algorithm2e? Can independent variables with low correlation with dependent variable be significant predictors? I have eight independent variables and one dependent.  I have run a correlation matrix, and 5 of them have a low correlation with the DV.  I have then run a stepwise multiple regression to see whether any/all of the IVs can predict the DV.  The regression showed that only two IVs can predict the DV (can only account for about 20% of the variance though), and SPSS removed the rest from the model.  My supervisor reckons that I have not run the regression correctly, as due to the strength of the correlations, I should have found more predictors in the regression model. But the correlations were tiny, so my question is: if IVs and the DV hardly correlate, can IVs still be good predictors of the DV? I have seen some hypotheses that the recent series corporate actions and statements are a sign of SO/SE being in the process of being, or about to be, sold. Arguments cited include the corp statements being made in paywalled areas and addressed to an audience that seems very distinct from the community, the choice of CEO, the push for a 'service sector' flavour of tensely smiling hospitality, and surely some more I can't recall in a coherent form.   Just how plausible and implausible, and how founded or unfounded, is such a hypothesis? Is it, as far as people more knowledgeable either about business in general or about the specific case of SE/SO, strategically consistent or even outright connected with the corp's behaviour (and a likely cause thereof)?