Terry Tao gives this oblique definition of **quasirandom** group in his [notes 3](https://terrytao.wordpress.com/2011/12/16/254b-notes-3-quasirandom-groups-expansion-and-selbergs-316-theorem/)

> $G$ is quasi-random (of order $D$) if all non-trivial unitary representations $\rho: G \to U(H)$  have dimension at least $D$.

There is nothing random about this representation, and the size of these irreducible representations doesn't make it more random.

There is someting kind of mixing about it, I guess:

> Let $G$ be a finite $D$ quasi-random group and $f,g \in \ell^2(G)$ such that $\mathbb{E}[f] = \mathbb{E}[g] = 0$ then $$ ||f \ast g||_{\ell^2(G)} \leq  \frac{1}{\sqrt{D}} \; |G| \;  \big|\big|f\big|\big|_{\ell^2(G)} \; \big|\big|g\big|\big|_{\ell^2(G)}  $$

Not knowing any better, I'd say this looks somewhat like the Cauchy-Schwartz inequality and I'd expect to use it in the same way.

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I don't know what it means to say a group $(G, \cdot)$ is quasi-random.  Are there any combinatorial objects which are "quasi-random" associated to $G$ ?  

> In the case of $SL(2, \mathbb{F}_p)$ we find it is $\frac{p-1}{2}$-quasirandom.  Can you randomly generate objects using the group $SL(2, \mathbb{F}_p$ ?