1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


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

    and you said 5 because....
    (I can not find anything to do with order of maps in my book)
  5. Nov 24, 2008 #4


    User Avatar
    Staff Emeritus
    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.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Automorphisms and Order of a Map