I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation.
That is, a finite group $G$ is a Frobenius complement if and only if there is a finite dimensional complex representation $\rho:G\to GL(V)$ so that the linear mapping $\rho(g)$ has no nonzero fixed points for any $g\in G\setminus\{1\}$.
**Is there a citable reference for this fact?**

I have found references that mention that this is well-known. For example:

It is well-known that a finite group $H$ has the structure of a Frobenius complement if and only if it can act fixed-point-freely on some finite group $G$ (this means that non-trivial elements of $H$ fix only the identitty element of $G$). Equivalently, we may require that $H$ acts fixed-point-freely on some elementary abelian group $E$, or on a linear space $V$ over a field of characteristic zero. (Aner Shalev: A new characterization of frobenius complements)

I'm aware of Passman's book *Permutation groups*, but it does not state this characterization.
I would prefer a reference that states this characterization explicitly (and preferably includes a proof).
My intended audience does not have a strong background in representation theory, so references of a more implicit kind ("this can be inferred from various results in this book" or "it is folklore/well-known/true that…") would be inconvenient.

**Edit:**
The references and descriptions given in the comments and the answer are very useful but do not contain an explicit reference for the representation theoretical characterization of Frobenius complements I'm looking for.
The direction that Frobenius complements have fixed-point-free representations is given as Theorem 18.1v in Passman's book, but the other direction is still missing.