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Homework Help: Automorphisms and Order of a Map

  1. Nov 24, 2008 #1
    1. The problem statement, all variables and given/known data
    Let G={(a,b)/ a,b[tex]\in[/tex]Z} be a group with addition defined by (a,b)+(c,d)=(a+c,b+d).
    a) Show that the map[tex]\phi[/tex]:G[tex]\rightarrow[/tex]G defined by [tex]\phi[/tex]((a,b))=(-b,a) is an automorphism of G.

    b) Determine the order of [tex]\phi[/tex].

    c) determine all (a,b)[tex]\in[/tex]G with [tex]\phi[/tex]((a,b))=(b,a)


    2. Relevant equations



    3. The attempt at a solution
    a) multiple parts
    1) Let (a,b)[tex]\in[/tex]G, then [tex]\phi[/tex](a,b)=(-b.a), which is an element in G since -b,a[tex]\in[/tex]Z therefore [tex]\phi[/tex](a,b)=(-b,a)[tex]\in[/tex]G
    2) One to One: suppose [tex]\phi[/tex](a,b)=[tex]\phi[/tex](c,d). That implies (-b,a)=(-d,c), which implies a=c and b=d therefore one to one.
    3) Onto: Let (c,d)[tex]\in[/tex]G to find (X,X)[tex]\in[/tex]G such that [tex]\phi[/tex](X,X)=(c,d)=[tex]\phi[/tex]((d,-c)) therefore onto
    4) Operation Preserving: Let (a,b),(c,d)[tex]\in[/tex]G. [tex]\phi[/tex]((a+c,b+d))=(-(b+d),a+c) and [tex]\phi[/tex]((a,b))+[tex]\phi[/tex]((c,d))=(-b,a)+(-d,c)=(-b-d,a+c)=(-(b+d),a+c)
    therefore [tex]\phi[/tex] is an isomophism, and since [tex]\phi[/tex] is from G to G, by definition [tex]\phi[/tex] is an automorphism of G

    b) I am not sure where to start for the order of a map?

    c) the set {(a,0)/a[tex]\in[/tex]G} thus [tex]\phi[/tex]((a,0))=(-0,a)=(0,a)
     
  2. jcsd
  3. Nov 24, 2008 #2
    for part (ii) the order of [tex] \phi[/tex] is 5. (at least from my world over here, it looks like that).
    I assume that the order of a mapping would be, whenever:

    [tex]\phi^k=\phi[/tex] so looking at your problem

    [tex]\phi^5(a,b)=\phi(a,b)=(-b,a)[/tex]

    Check the calculations, i did it really fast....
     
  4. Nov 24, 2008 #3
    are the other two parts ok?

    and you said 5 because....
    [tex]\phi[/tex]((a,b))=(-b,a)
    [tex]\phi[/tex]((-b,a))=(-a,-b)
    [tex]\phi[/tex]((-a,-b))=(b,-a)
    [tex]\phi[/tex]((b,-a))=(a,b)
    [tex]\phi[/tex]((a,b))=(-b,a)
    right?
    (I can not find anything to do with order of maps in my book)
     
  5. Nov 24, 2008 #4

    HallsofIvy

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    Science Advisor

    Yes, the first two parts are ok.

    The "order of a map" is, as sutupidmath said, the smallest positive integer, k, such that [itex]\phi^k= \phi[/itex]. Did you look up "order" in the index of your book?
     
  6. Nov 24, 2008 #5
    Yea, but I wasnt sure if I could apply the same rules to a map. In my book it just has for order is of a group or the order on an element.

    Thanks for your help and for clarifing that.
     
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