1. Limited time only! Sign up for a free 30min personal 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!

Isomorphism! Help!

  1. Oct 28, 2008 #1
    The question is this: How many isomorphisms f are there from G to G' if G and G' are cyclic groups of order 8?

    My thoughts:

    Since f is an isomorphism, we know that it prserves the identity, so f:e-->e', e identity in G, e' identity in G'.

    Also f preserves the order of each element. That is if o(a)=k=>o(f(a))=k

    SO, i thought that f will send the el of the same order in G to the corresponding elements of the same order in G'.

    Let G=[a], and G'=. so it means that there are 4 el in G that have order 8 ( the generators of G, a, a^3, a^5,a^7), so there are 4 possibilities for these elements, hense by keeping the other el. fixed we would have 4^4 isomorphisms.

    But also we have 2 el of order 4, (a^2, and a^6) so there are two possibilities for these elements to be mapped into G' by f, so if the other el are fixed we would have 2^2 mappings.

    Does this mean that the total nr of such isomorphisms is 4^4+2^2????? Or am i totally on the wrong way???

    Any suggestions would be appreciated.
  2. jcsd
  3. Oct 28, 2008 #2


    User Avatar

    f must satisfy f(a)=bk, some 1 <= k <= 7, and f is completely determined by the choice of k.
    You should be able to see how many possible choices there are for k.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Isomorphism Help Date
Isomorphisms abelian help Aug 20, 2014
Isomorphism problem and need help Oct 31, 2009
Isomorphism homework help Sep 25, 2008
Isomorphism help Sep 13, 2008
Isomorphic Help! May 23, 2007