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> Find all homomorphisms from $D\_4$ to $S\_3$.
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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: $f(x\_i)^2 = e = f(x\_i) f(x\_j)$.