Let $G=(Z_4,+)$ and let $H=(U_5,*)$ where $U_5 = \{[1],[2],[3],[4] \}$ . I know that $[1]$ and $[3]$ are both generators for $G$. I also know that $[2]$ and $[3]$ are both generators for $H$. In order to construct isomorphism we have to map a generator to a generator. And so, $\phi([1]^x_4) = [2]^x_5$ is an isomorphism from $G \longrightarrow H$. Also $\phi([1]^x_4) = [3]^x_5$ is another isomorphism from $G \longrightarrow H$.  Can we do the same thing for the other generator of $G$. And so we would have two more isomorphisms, Namely $\phi([3]^x_4) = [2]^x_5$ and $\phi([3]^x_4) = [3]^x_5$  is that true ? and if it is true, then does that mean that if we have $n$ generators for a group $G$ and $m$ generators of a group $H$ then i would have $nm$ isomorphisms in total ? Thanks
So I am asked to find all the isomorphisms between $G = (\mathbb Z_4,+)$ and $H = (U_5,*)$.  I solved it as follows: we will have two isomorphism corresponding to the two generators of $U_5$.  The first isomorphism : $\phi([x]_4)$ = $[2]_5^x$   The second isomorphism is: $\phi([x]_4)$ = $[3]_5^x$   Notice that [2] and [3] are the only generators of $U_5$ one can check it systematically or use the fact that $a^m$ is a generator iff gcd(m,4) = 1.  So I just wanted a verification of my work I showed that they are bijective, well defined, and is a homomorphism.
Moving linux install to a new computer I know that it can, in some circumstances, be difficult to move a Windows installation from one computer to another (physically move the hard drive), but how does that work on linux? Aren't most of the driver modules loaded at bootup? So theoretically would it be that much of a hassle?  Obviously xorg configs would change and proprietary ATI drivers and such would have to be recompiled (maybe?). Is there more to it than I'm thinking of?  Assume the 2 computers are from the same era, ie both i7s but slightly different hardware.  Update: Thanks for the answers. This is mostly for my own curiosity. I have my linux system up and running at work, but eventually I'd like to move to a computer that I can get dual video cards into so I can run more than 2 monitors. But not any time soon How can friction do no work in case of pure rolling? I have read various answers , on PSE and elsewhere , and most of them explain that the point of contact of the rolling object undergoes 0 instantaneous displacement in the direction of friction, I agree , but then there is a feeling that the friction does provide a torque , so how can it do 0 work . ( The pure rolling here is under a constant external force on the object so friction does act ). Find the slope of tangent line to the curve at the point $(1, \pi/2)$  the equation is  $$\sin(xy) = x$$  The right answer was = Slope is infinite   My answer was taking the derivative and I end up in this equation   $$ (1-y\cos(xy))/ x\cos(xy)  $$  I substitute with the points and the calculator showed an error, Does the error means infinite ? What do you call someone who always puts blame on others? How to convert a 3 column csv file into a table (or matrix) I have an CSV input file format like this, with a nucleotide sequence in field 1, text in field 2, and an integer in field 4:  ATGC,CD3,56 ATGC,CD4,67 ATGC,IgD,126 ATGC,IgM,127 AGTC,CD3,67 AGTC,CD4,78 AGTC,IgD,102 AGTC,IgM,89 TCGA,CD3,334 TCGA,CD4,123 TCGA,IgD,456 TCGA,IgM,80 CGTA,CD3,54 CGTA,CD4,32 CGTA,IgD,82 CGTA,IgM,117   I opened this CSV file using Numbers in Mac it's display as 3 columns format, however, I want to convert it to the table(or Matrix) format (also a CSV file), making the first column, the nucleotide sequences, into a header, and want the result to also look like a table (or Matrix):       ATGC  AGTC  TCGA  CGTA CD3  56    67    334   54 CD4  67    78    123   32 IgD  126   102   456   82 IgM  127   89    80    117   Below is a section from my real input CSV file (sample input.txt):  AGAATAGTCTGATTCT,-,,38 AGAATAGTCTGATTCT,AnnexinV,,51 AGAATAGTCTGATTCT,CD127,,39 AGAATAGTCTGATTCT,CD138,,3 AGAATAGTCTGATTCT,CD14,,2 AGAATAGTCTGATTCT,CD16,,4 AGAATAGTCTGATTCT,CD19,,10 AGAATAGTCTGATTCT,CD20,,6 AGAATAGTCTGATTCT,CD24,,21 AGAATAGTCTGATTCT,CD25,,4 AGAATAGTCTGATTCT,CD27,,87 AGAATAGTCTGATTCT,CD3,,235 AGAATAGTCTGATTCT,CD34,,5 AGAATAGTCTGATTCT,CD38,,18 AGAATAGTCTGATTCT,CD4,,412 AGAATAGTCTGATTCT,CD43,,99 AGAATAGTCTGATTCT,CD5,,430 AGAATAGTCTGATTCT,CD56,,3 AGAATAGTCTGATTCT,CD8,,7 AGAATAGTCTGATTCT,IgD,,4 AGAATAGTCTGATTCT,IgM,,2 TGTGGTAGTTCGTCTC,-,,9 TGTGGTAGTTCGTCTC,AnnexinV,,42 TGTGGTAGTTCGTCTC,CD127,,6 TGTGGTAGTTCGTCTC,CD138,,4 TGTGGTAGTTCGTCTC,CD16,,40 TGTGGTAGTTCGTCTC,CD19,,7 TGTGGTAGTTCGTCTC,CD20,,2 TGTGGTAGTTCGTCTC,CD24,,24 TGTGGTAGTTCGTCTC,CD25,,2   How can I do this using Linux text formatting commands? Whoever or whomever: 'happy for ___ has the pleasure of working with you next.' So sad to lose you, yet happy for whomever has the pleasure of working with you next. Should questions where a problem arose from a typo be closed? Should questions where a problem arose from a typo be closed?  Here is a recent example:    The problem was a misspelled _gaq (written as _gat).   Should these be closed? Surely they're of no use to anyone once the OP recognizes the mistake. If so, what justification should be given? Not a real question? Too localized?     Just to be clear, the question linked above actually was not broken due to a typo. This was my mistake, and raises some other interesting questions. See the discussion below.  If anyone feels like editing this question with examples of actual typo questions, go for it. Every Function in a Finite Field is a Polynomial Function When signing in with Google (OAuth), it shows a list of accounts it already knows about. How can I remove some accounts from this list?    I know I can nuke all of them by deleting all cookies, but I'd prefer to remove one account that I don't want there, rather than all of them.  There used to be an ability to edit this list in the older version of this screen, but in 2018 redesign Google has removed [×] buttons next to the accounts. I was going through fluid dynamics and I came upon this section on the shape of a liquid surface in a rotating cylindrical vessel. The working in the book is something like this:   It considers a mass $m$ of water on the surface of the rotating liquid at a distance $x$ from its axis of rotation .It takes into account a pseudo force ($m \cdot x \cdot \omega^2$) radially outwards and its weight force ($mg$).It then proceeds to calculate the net force and the angle it makes with the vertical.From there through basic integration it gets the equation of the surface of liquid  i.e. $y= \dfrac{(x^2 \cdot \omega^2)}{2g}$ .  I get this working.  But if I were to view the same rotation through an inertial frame,how would I proceed? I mean, there wouldn't be any pseudo force, rather a centripetal acceleration of same magnitude radially inwards.This is where I am facing certain doubts.  The forces on the liquid would be  the centripetal force inwards and the weight force downwards.The net force would therefore be in a different direction,and consequently the slope of the surface at that point would be in a different direction.How is it possible for the same surface to have two different shapes in two different frames?   I would be obliged if someone could help me out by pointing out the flaw in my understanding. I'm trying to do raster algebra with two rasters, but when I transform the rasters to array I can't do the operation because the files do not have the same extent  example:  from osgeo import gdal import numpy as np  IMG1 = gdal.Open('C:/IMG1.TIF') IMG2 = gdal.Open('C:/IMG2.TIF')  img1 = IMG1.ReadAsArray() img2 = IMG2.ReadAsArray()  img1+img2 &gt;&gt;&gt;&gt;&gt; ValueError: operands could not be broadcast together with shapes (5353,3352) (5548,3232)    So.. if I see the gt of every img show that img1 and img2 have the same size of the pixel (100) but different extent  g1 = IMG1.GetGeoTransform() gt1 &gt;&gt;&gt;&gt;&gt; (482084.999999999, 100.0, 0.0, 5652715.00001374, 0.0, -100.0) g2 = IMG2.GetGeoTransform() gt2 &gt;&gt;&gt;&gt;&gt; (475184.999999999, 100.0, 0.0, 5643715.00001535, 0.0, -100.0)   So.. I used this operation in order to get the intersect extent.. but I don't know how to apply  r1 = [gt1[0], gt1[3],gt1[0] + (gt1[1] * IMG1.RasterXSize), gt1[3] + (gt1[5] * IMG1.RasterYSize)] r2 = [gt2[0], gt2[3],gt2[0] + (gt2[1] * IMG2.RasterXSize), gt2[3] + (gt2[5] * IMG2.RasterYSize)] intersection = [max(r1[0], r2[0]), min(r1[1], r2[1]), min(r1[2], r2[2]), max(r1[3], r2[3])] Can I transfer my Terraria character from one PC to another? Currently my map server is run on my laptop at home, and my brother and I play via LAN each on his desktop PC.  However, I'd like to transfer my character from the Desktop PC to my laptop in order to play on the go.  Can I do this? Where is the character information saved? When is the past perfect exactly needed? I have never had this problem before. But after seeing some questions posted on ELU, I started to doubt. I have been through links1, pages of grammar books2, a lot of questions3 posted here and ELU. I now still can't seem to understand how the past perfect is exactly used. The references sometimes even contradict each other.  What I had understood so far was      The past perfect is not necessary if we are not 'going back' to the earlier past, and simply moving from one event to another.   but I then saw a lot of other 'rules' including      If the sequence is clear, the past perfect is not needed.   which is slightly different, less 'narrow' and can be applied in more situations than the first ruleA, and      The past perfect is optional only when talking about an action at a specific time.   This is confusing. Can anyone here clarify things out please?    A For example, for  Doing the survey [completed] -&gt; Showing the result      We showed the result of the survey we did.      If the second rule is applied, the sentence above is fine (the sequence is clear), but not if the first rule since we are not moving from any event to the next and possibly even going back.    1  and   2 Practical English Usage (2005) by Michael Swan, Collins Cobuild Student's Grammar (1991). Upper-Intermediate Cutting Edge.  3 including , , ,  etc. Time travel film: character with male and female reproductive organs is taken back in time by a barman who turns out to be themselves I have had a friend give me a rough description of a film and I'm curious to find out what it actually is. I don't know very much. I understand this will be difficult, especially since every time my friend explains the tale it's different so this description is likely to contain some inaccuracies.  The film's plot begins with a recounting given by the main character, who had both male and female reproductive organs but had one set removed (not sure which, think male ones were removed). The main character is narrating this tale to the barman in a bar. The barman then takes the main character back in time, where they make love to themselves and produce a child.  The child, once born, is abandoned by the mother in the past - but the barman takes the child to the orphanage and leaves them there, ensuring their survival. Eventually it is revealed that the child was the main character themselves, and so was the barman.  In other words every major character introduced up to that point was the same person who had severely messed with their own timeline. I had studied group theory a year ago, but still could not understand the proof involving The Correspondence theorem.  let $G$ be a group and let $N⊴G$, where  $N⊴G$ indicates that $N$ is a normal subgroup of $G$. Then there is a bijection from the set of subgroups $A$ of $G$ that contain $N$ onto the set of subgroups $\overline A $=$A/N$ of $G/N.$  Can anyone explain the steps to me in:  $(1.)$ a very easy-going manner so that I understand importance of each step. $(2.)$I don't require the proof but only how the step follow each other .  I'll be thankful if there is anyone kind enough to do this.... If $x$ and $y$ are not both $0$ then $ x^2 +xy +y^2> 0$ Real life uses of Moment generating functions In most basic probability theory courses your told moment generating functions (m.g.f) are useful for calculating the moments of a random variable. In particular the expectation and variance. Now in most courses the examples they provide for expectation and variance can be solved analytically using the definitions.   Are there any real life examples of distributions where finding the expectation and variance is hard to do analytically and so the use of m.g.f's was needed? I'm asking because I feel like I don't get to see exactly why they are important in the basic courses. Autosegmental Representation in tikz-qtree This is what I want to achieve    and this:    But all I've been able to make with tikz-qtree is the following:  \documentclass[12pt,a4]{article} \usepackage[utf8]{inputenc} \usepackage{tipa,tikz,tikz-qtree} \begin{document}  \begin{tikzpicture} [baseline] \tikzset{frontier/.style={distance from root=90pt}} \Tree [.$\sigma$ [.O [ p ] ] [.R [.$\mu$ a ] [.$\mu$ : ] ] ] \end{tikzpicture} \begin{tikzpicture} [baseline] \tikzset{frontier/.style={distance from root=90pt}} \Tree [.$\sigma$ [.O [ p ] ] [.R [.$\mu$ a ]  ] ] \end{tikzpicture}   And this:  \begin{tikzpicture}[baseline] \tikzset{frontier/.style={distance from root=90pt}} \Tree [.$\sigma$ [.O [ l ] ] [.R [.$\mu$ a ] [.$\mu$ l ] ] ] \end{tikzpicture} \begin{tikzpicture} [baseline] \tikzset{frontier/.style={distance from root=90pt}} \Tree [.$\sigma$ [.O [ : ] ] [.R [.$\mu$ a ] [ l ] ] ] \end{tikzpicture} \end{document}   Which renders:    and    The important part here is the linking from the leaves to more than one parent (it would be nice if I had the option of making the line dashed too). Now, pst-asr doesn't quite achieve what I want it to. It's important that I have the moras ('mu's). (Also it's a pstricks package, which means I must typeset it in DVIPSPDF.) I have tried a bit of the forest-package (especially forest-GP1), but I can't make that work as I want either. Lastly, some have recommended xyling, but I find that package very difficult to use.  Any recommendations?  Thanks. which app triggers the "Windows features" dialog? I keep getting this window titles "Windows Features" It then says that:     An app on your PC needs the following Windows feature:      .NET Framework 3.5 (Includes .NET 2.0 and 3.0)   and I can choose to install or skip (being warned that my apps might not work correctly).  There is also a "telll me more about this feature" link but it explains about the .NET framework.  I want to know if there is a way for me to know which application is it that asks for this feature as the window just keeps popping out of the context of activating a new app...  This all started just a few days ago after some unattended update on my windows 10. For symmetric stable distributions, why is $\alpha \le 2$?