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Math Help - Infinite Dihedral group

  1. #1
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    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  D_{\infty} be a group with presentation  <a,b\mid a^2=b^2=e>

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

    Hint: Find a short exact sequence which splits. The sequence should be clear from the isomorphism.
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  2. #2
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    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  D_{\infty} be a group with presentation  <a,b\mid a^2=b^2=e>

    Show that  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 D_{\infty} is in the form (ab)^na^k, where n \in \mathbb{Z}, \ k \in \{0,1\}. now define f: \mathbb{Z}/2 \longrightarrow D_{\infty} by f([k])=a^k, and

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

    1 \longrightarrow \mathbb{Z}/2 \longrightarrow D_{\infty} \longrightarrow \mathbb{Z} \longrightarrow 1 is exact, split and hence D_{\infty} \simeq \mathbb{Z} \rtimes \mathbb{Z}/2.
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