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?

Answer 2

Suppose that $\theta : D_4 \to S_3$ is a homomorphism. Then, by the first isomorphism theorem:

• $K=\ker \theta \unlhd D_4$
• $\theta(D_4) \leq S_3$
• $\dfrac{|D_4|}{|K|} = |\theta(D_4)|$

So the associated subgroups that arise from the possible homomorphisms must each satisfy these 3 relations, restricting us to a handful of case.

By Lagrange, any subgroup of $S_3$ must have order $1,2,3$ or $6$; and that $|K| =1,2,4,8$. So, by our third condition, this immediately restricts us to $(|K|,|\theta(D_4)|) \in \{(4,2), (8,1)\}$.

Observe that the homomorphism leading to the $(8,1)$ pairing above must have kernel $D_4$ i.e. must map everything to the identity permutation in $S_3$ - there is only one such function and it is indeed a homomorphism.

For the $(4,2)$ pairing, it's easily seen that the only order 2 subgroups of $S_3$ are those generated by each of the transpositions $\tau$, say. Since knowing where the generators map to tells us everything about the homomorphism, as $\theta$ preserves group structure, it suffices to break this into cases:

$ \underline{(1) \ \theta(s)=\tau:}$ It remains to consider where $r$ maps to under $\theta$, this breaks up into two sub-cases:

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (a)\ \theta (r ) = e $ - Only have the three homomorphisms s.t. $r^i\mapsto e, sr^i \mapsto \tau$ for each $\tau \in S_3$

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (b) \ \theta(r ) = \tau$ - Then have the maps s.t. $r^{2i+1}, sr^{2i} \mapsto \tau; r^{2i}, sr^{2i+1} \mapsto e$ for each $\tau \in S_3$

$\underline{(2) \ \theta(s) = e:}$ Considering $\theta (r )$ as above:

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (a)\ \theta (r ) = e $ - Trivial map, which we have already accounted for above.

$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (b)\ \theta (r ) =\tau $ - Have maps s.t. $r^{2i+1}, sr^{2i+1} \mapsto \tau; r^{2i}, sr^{2i}\mapsto e$ for each $\tau \in S_3$

Yielding 10 distinct homomorphisms $D_4 \to S_3$.