The infinite dihedral group $D_\infty$ is a subgroup of permutations of the integers generated by $f(n) = -n$ and $g(n) = 1-n$, which reflect the integer number line over the point 0 and 1/2 respectively.
(a) Give a presentation of $D_\infty$.
(b) Demonstrate a surjective homomorphism to each finite dihedral group $\varphi:D_\infty\to D_n$ for $n \geq 3$
For part (a) I have the presentation $<x,y \space|\space x^2=y^2=1>$
Given my presentation how to I demonstrate a surjective homomorphism to any finite dihedral group.
any help is appreciated.
Best Answer
This looks like homework so I won't directly answer your question, but I'll describe some theory which will hopefully be helpful.
Firstly, suppose $G = \langle g_1, \ldots, g_n|r_1, \ldots, r_m\rangle$, where the $g_i$ are generators and the $r_j$ are relations. What does this really mean? It's the quotient group $$G = \frac{F(g_1, \ldots, g_n)}{N(r_1, \ldots, r_m)},$$ where I've written $F(g_1, \ldots, g_n)$ to denote the free group generated by the $g_i$, and I've written $N(r_1, \ldots, r_m)$ to denote the normal closure of the $r_i$ (that is, the smallest normal subgroup of $F(g_1, \ldots, g_n)$ containing each relation $r_i$).
But on the other hand, we know that homomorphisms mapping from free groups are uniquely determined by where they send the generators. For example, if I want to define a homomorphism $f: \langle x, y \rangle \to G$, I just have to say where $x$ and $y$ map to, and the rest of the homomorphism is determined.
In part (b) of your question, you want to define homomorphisms using presentations. By thinking of the domain and codomain as quotient groups, hopefully you can think about how to define a surjective homomorphism. The main thing to be careful about is that the homomorphism is well defined, i.e. anything in the normal closure of the relations in the domain really maps to zero.