1. ## 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.

2. Originally Posted by bulls6x
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 $\displaystyle D_{\infty} \simeq \mathbb{Z} \rtimes \mathbb{Z}/2.$