[Math] Homomorphisms from $D_4$ to $S_3$.

Question

Find all homomorphisms from $D_4$ to $S_3$.

We have $D_4 = \{e,r,r^2,r^3,s,sr,sr^2,sr^3\}$ (where $r^4 = e = s^2$) and $S_3 = \{e,(12),(13),(23),(123),(132)\} = \langle (12) (13) \rangle$.

Let $x_i = (1i) \in S_3$, where $i \in \{2,3\}$. By another exercise (already proved in my class), the following relation holds: $x_i^2=e=(x_i x_j)^3$.

Mapping the relation onto its image, I get $f(x_i^2)=f(e)=f(x_i x_j)$, and of course $f(e)=e$. But we are dealing with homomorphisms, so this is also true: $f(x_i)^2 = e = f(x_i) f(x_j)$.

Answer 1

For a homomorphism $$\eta : D_8(\langle r,s\rangle ; r^4=s^2=e;rs=sr^3)\rightarrow S_3(\langle(12),(13)\rangle)$$

$a^n=e\Rightarrow \eta(a^n)=e\Rightarrow (\eta(a))^n=e$

Possible images of elements of order $2$ is an element of order $2$

Possible images of elements of order $4$ is an element of order $4$

• $\eta(r)\in \{??\}$
• $\eta(r^2)\in \{(12),(13),(23)\}$
• $\eta(r^3)\in \{??\}$
• $\eta(s)\in \{(12),(13),(23)\}$
• $\eta(rs)\in\{??\}$
• $\eta(r^2s)\in\{??\}$
• $\eta(r^3s)\in\{??\}$

How many possible maps are there?