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!

Can anyone help me with these abstract algebra proofs?

  1. Oct 30, 2012 #1
    1. The problem statement, all variables and given/known data

    a) Let [itex]H[/itex] be a normal subgroup of [itex]G[/itex]. If the index of [itex]H[/itex] in [itex]G[/itex] is [itex]n[/itex], show that [itex]y^n \in H[/itex] for all [itex]y \in G[/itex].

    b) Let [itex]\varphi : G \rightarrow G'[/itex] be a homomorphism and suppose that [itex]x \in G[/itex] has order [itex]n[/itex]. Prove that the order of [itex]\varphi(x)[/itex] (in the group [itex]G'[/itex]) divides [itex]n[/itex]. (Suggestion: Use the Division Algorithm.)

    c) Let [itex]\varphi : \mathbb{Z}_n \rightarrow \mathbb{Z}_m[/itex] be a homomorphism. Show that [itex]\varphi[/itex] has the form [itex]\varphi([x]) = [qx][/itex] for some 0 ≤ [itex]q[/itex] ≤ [itex]m[/itex] - 1. Then, by means of a counterexample, show that not every mapping from [itex]\mathbb{Z}_n[/itex] to [itex]\mathbb{Z}_m[/itex] of the form
    [itex]\varphi([x]) = [qx][/itex] where 0 ≤ [itex]q[/itex] ≤ [itex]m[/itex] - 1 need be a homomorphism.

    2. Relevant equations

    For normal subset H:

    [itex]yH=Hy[/itex] (right coset = left coset) for all [itex]y \in G[/itex], and they partition [itex]G[/itex].
    [itex]yhy^{-1} \in H[/itex] for all [itex]h \in H[/itex], [itex]y \in G[/itex].

    For homomorphism [itex]\varphi : G \rightarrow G'[/itex]:

    [itex]\varphi(ab) = \varphi(a) \varphi(b)[/itex] for all [itex]a,b \in G[/itex].

    3. The attempt at a solution

    b):

    [itex]x^n = e; n \in \mathbb{P}[/itex]
    [itex](\varphi(x))^{qn+r} = e; q,r \in \mathbb{Z},[/itex] 0≤ r < n.
    [itex](\varphi(x))^{qn}(\varphi(x))^{r}=e[/itex]
    [itex]\varphi(x^{qn})\varphi(x^r)=e[/itex]
    [itex]\varphi(e)\varphi(x^r)=e[/itex]
    [itex]\varphi(x^r)=e[/itex]
    ....? Not sure where to go from here.
     
    Last edited: Oct 30, 2012
  2. jcsd
  3. Oct 30, 2012 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    For (a), what can you say about the element yH of G/H ?
     
  4. Oct 30, 2012 #3
    Hmm... The order of yH = order of G divided by n...? That, and it contains y?
     
    Last edited: Oct 30, 2012
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Can anyone help me with these abstract algebra proofs?
  1. Can anyone help me? (Replies: 4)

Loading...