I'm struggling trying to resolve the following problem:
Determine whether the symmetric group $S_3$ and the quaternion group, of order 8, admit a complete mapping. If it is the case, build one.
I know that a complete mapping for a quasigroup $(Q, \cdot)$ is a bijective function $\theta:Q \longrightarrow Q$ such that the function $\eta:Q\longrightarrow Q$ defined by $\eta(q)=q \cdot \theta(q)$ is bijective.
I also know that a finite quasigroup $(Q, \cdot)$ has a complete mapping $\iff$ his table of multiplication has a transversal.
I read that $S_3$ has not a complete mapping, but I can't find a way to prove it, can someone help me?