Results 1 to 2 of 2

Thread: Infinite Dihedral group

  1. #1
    Newbie
    Joined
    Jan 2009
    From
    Austin, Tx
    Posts
    9

    Infinite Dihedral group

    Here's a problem I had last semester in my grad Algebra class which really helped me understand short exact sequences that split.

    Let $\displaystyle D_{\infty}$ be a group with presentation $\displaystyle <a,b\mid a^2=b^2=e> $

    Show that $\displaystyle G \simeq\mathbb{Z}\rtimes\mathbb{Z}/2$.

    Hint: Find a short exact sequence which splits. The sequence should be clear from the isomorphism.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by bulls6x View Post
    Here's a problem I had last semester in my grad Algebra class which really helped me understand short exact sequences that split.

    Let $\displaystyle D_{\infty}$ be a group with presentation $\displaystyle <a,b\mid a^2=b^2=e> $

    Show that $\displaystyle G \simeq\mathbb{Z}\rtimes\mathbb{Z}/2$.

    Hint: Find a short exact sequence which splits. The sequence should be clear from the isomorphism.
    no need for Hint! the question is straightforward: every element of $\displaystyle D_{\infty}$ is in the form $\displaystyle (ab)^na^k,$ where $\displaystyle n \in \mathbb{Z}, \ k \in \{0,1\}.$ now define $\displaystyle f: \mathbb{Z}/2 \longrightarrow D_{\infty}$ by $\displaystyle f([k])=a^k,$ and

    $\displaystyle g: D_{\infty} \longrightarrow \mathbb{Z}$ by $\displaystyle g((ab)^n a^k)=n.$ see that $\displaystyle f,g$ are homomorphisms and $\displaystyle \ker g=\text{im} f.$ also $\displaystyle h: \mathbb{Z} \longrightarrow D_{\infty}$ defined by $\displaystyle h(n)=(ab)^n$ satisfies $\displaystyle gh=\text{id}_{\mathbb{Z}}.$ therefore the sequence

    $\displaystyle 1 \longrightarrow \mathbb{Z}/2 \longrightarrow D_{\infty} \longrightarrow \mathbb{Z} \longrightarrow 1$ is exact, split and hence $\displaystyle D_{\infty} \simeq \mathbb{Z} \rtimes \mathbb{Z}/2.$
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Dihedral Group
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: Nov 22nd 2010, 03:27 PM
  2. infinite dihedral group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Feb 14th 2010, 10:12 PM
  3. Dihedral Group
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: Sep 30th 2009, 12:27 PM
  4. Dihedral group
    Posted in the Advanced Algebra Forum
    Replies: 7
    Last Post: Nov 12th 2008, 05:57 PM
  5. Group Theory Question, Dihedral Group
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: Mar 4th 2008, 10:36 AM

Search Tags


/mathhelpforum @mathhelpforum