Let $n$ and $p$ be positive integers. Is there a finite group $G_p$ generated by elements $a_1, \dots, a_n$ such that any nonempty reduced word on $a_1, \dots, a_n, a_1^{-1}, \dots, a_n^{-1}$ of size $\leq p$ is not equal to the identity element?
I know that the answer is yes (it follows from the residual finiteness of the free group of rank $n$). But I would like a more constructive solution. Ideally, I would like to know the examples of such $G_p$, that satisfy $\log |G_p| = O(p)$.