Explain why it is impossible for $G$ to have such Sylow $p$-subgroups Let $G$ be a finite group. Explain why it is impossible for $G$ to have one Sylow $2$-subgroup isomorphic to $\mathbb{Z}_4$ and another Sylow $2$-subgroup isomorphic to $\mathbb{Z}_2 \times \mathbb{Z}_2$.  My idea: By the second Sylow theorem we know that the Sylow $p$-subgroups are always conjugate to each other; and thus, they have the same order. So $\mathbb{Z}_4$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ must be conjugate. In other words,  $\mathbb{Z}_4$ and $\mathbb{Z}_2 \times \mathbb{Z}_2$ are isomorphic, wich is impossible. Is this correct?
Sylow Theorem Explanation The Sylow theorems say that all $p$-Sylow subgroups are conjugate.   Is that the only thing they are conjugate to? In other words could a $p$-Sylow subgroup also be conjugate to something that is not a $p$-Sylow group?   Or does the fact that every element with order dividing $p$ is in a $p$-Sylow group mean that that is never possible?
Why is the range of inverse trigonometric functions defined in this way? My question is really simple, why is the range of the $\sin^{-1}(x)$, $\cos^{-1}(x)$ and $\tan^{-1}(x)$ defined as $[-\pi/2,\pi/2]$, $[0,\pi]$ and $[-\pi/2,\pi/2]$ respectively? Is there some particular reason? We could choose another range for each inverse trigonometric function. For example, we can pick $[0,\pi]$ to be the range of $\sin^{-1}x$.  EDIT  I've understood why the range of the sine and cosine has to be $[-\pi/2,\pi/2]$ and $[0,\pi]$ respectively. I'm still wondering why can't we define the range of the tangent as $[0,\pi]$ What is the Hausdorff dimension of the set of irrationals on $\mathbb R^1$ I am sorry if this is a dump question but I can't find any reference for this. What is the Hausdorff dimension of the set of irrationals on $\mathbb R^1$? I speculate this is $1$ but don't know how to prove it rigorously. Error AL lib: UpdateDeviceParams: Failed to set 44100hz, got 48000hz instead. What can I do? I have Win 10 and when I open Blender I have the error:     AL lib: UpdateDeviceParams: Failed to set 44100hz, got 48000hz instead Show that $\int_0^ \infty \frac{1}{1+x^n} dx= \frac{ \pi /n}{\sin(\pi /n)}$ , where $n$ is a positive integer. SQL query or method to list all sql services running This might be a simple one, but i searched a lot and unable to find one.  Is there a sql query or a way i can run on registered server to find the list of all sql related services running like, ssis , ssas, ssrs, sql, agent, broswer, etc...  Tried googling out and can find sys.dm_server_services dmv but it does not show all the services running I'm training a neural network but the training loss doesn't decrease. How can I fix this?  I'm not asking about overfitting or regularization. I'm asking about how to solve the problem where my network's performance doesn't improve on the training set.    This question is intentionally general so that other questions about how to train a neural network can be closed as a duplicate of this one, with the attitude that "if you give a man a fish you feed him for a day, but if you teach a man to fish, you can feed him for the rest of his life." See this Meta thread for a discussion:   If your neural network does not generalize well, see: I have a complex command that I'd like to make a shell/bash script of.  I can write it in terms of $1 easily:  foo $1 args -o $1.ext   I want to be able to pass multiple input names to the script. What's the right way to do it?    And, of course, I want to handle filenames with spaces in them. Is there a human home planet? Did all the humans in the Star Wars universe originate on a single planet some time in the distant past, or did they originate on different worlds and intermix into one species over millenia?  This is specifically about only humans, not humanoid aliens. 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Here's the document:  &lt;ArrayOfAnyType xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns="http://tempuri.org/"&gt;   &lt;anyType xsi:type="Document"&gt;     &lt;Id&gt;5&lt;/Id&gt;     &lt;Title&gt;T1&lt;/Title&gt;   &lt;/anyType&gt;    &lt;anyType xsi:type="Document"&gt;     &lt;Id&gt;15&lt;/Id&gt;     &lt;Title&gt;T15&lt;/Title&gt;   &lt;/anyType&gt; &lt;/ArrayOfAnyType&gt;   What's the XPath expression going to be if I want to extract all "anyType" elements with xsi:type="Document"?  I've tried this:   //anyType[@xsi:type="Document"]   and it doesn't work: Why cannot fermions have non-zero vacuum expectation value? Is $\int \frac{1}{2x} \, dx $ equal to $\frac{\ln|2x|}{2}+ C$ or $\frac{\ln|x|}{2}+ C$ Reasoning for $\frac{\ln|x|}{2}+ C$ :  $$\int  \frac{1}{(2x)}  \, dx = \frac{\ln|2x|}{2}+ C$$   Check: By taking derivative we get $\frac{1}{x}$  Reasoning for $\frac{\ln|x|}{2}+ C$ :  $$\int  \frac{1}{2x}  \, dx = \frac{1}{2}\int  \frac{1}{x}  \, dx =\frac{\ln|x|}{2}+ C$$  Check: By taking derivative we get $\frac{1}{x}$  Question: Is  $\int  \frac{1}{2x}  \, dx $ equal to $\frac{\ln|2x|}{2}+ C$ or $\frac{\ln|x|}{2}+ C$? Difference between deportation and removal As I understand it, there are two mechanisms for throwing somebody out of the UK: deportation and removal. Informally, people seem to call both of these "deportation".  What are the differences between them? Who do they apply to? Do they have different effects on the person's ability to return to the UK later? Contacts in Mavericks Messages display phone numbers instead of contact names Since updating to Mavericks the Messages app doesn't display the contact names for people I've been communicating with iMessage, it shows their phone number instead. My messages thankfully still stay synced between my iPhone and Messages on Mavericks. How to get a list of all the labels in a LaTeX document in WinEdt I am writing a large document using LaTeX and WinEdt.  I have labeled each chapter, section and so on.  I would like to print out a list of just the labels (so that I don't duplicate one later on, and so that I have it handy in order to write \refs).    Is there a way to do this? Explanation behind Second Derivative of a Parametric Equation Formula I am looking to find out how the second derivative formula works.  I can blindly apply it but I don't have a grasp of what is going on or why. The first derivitive has the formula $\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$. I told this was because if you divide them you cancel out the dts and get dy/dx. The second derivative has the formula $\frac{d^2y}{dx^2} = \frac{d}{dx} \left(\frac{dy}{dx}\right) = \frac{ \frac{d}{dt}\left(\frac{dy}{dx}\right) }{\frac{dx}{dt}}$. Is this something that can be explained with only a knowledge of calculus or is this something I would have to take on faith until I learn more. Any help would be greatly appreciated. Hi is it possible to get images from go daddy  to visual force pages.if it possible can you explain clearly how the process is.