Let $q$ denote a prime power and $\text{GL}_n(q)$ and $\text{U}_n(q^2)$ the general linear and unitary group, respectively. Then $\text{U}_n(q^2)$ is naturally a subgroup of $\text{GL}_{n}(q^2)$, so one kind of groups can be embedded into the other. Let $C(g)$ be the conjugacy class of an element $g$ in its respective group. Then we can define the *length* of $g$ to be $\ell(g):=\frac{\log|C(g)|}{\log|G|}$. The above embedding only changes the length by a constant factor.

Note that the length function induces a biinvariant metric by $d(g,h):=\ell(gh^{-1}$.

My question is if there are any functions $f,g:\mathbb{N}\rightarrow \mathbb{N}$ such that we can always find an embedding of $\text{GL}_n(q)$ into $\text{U}_{f(n)}(q^{g(n)})$, which doesn't change the above length function to much. It would be nicest if the distortion of length would only be a constant factor.

My feeling is that this is not possible, but I somehow fail to find an explanation.

A further question is what happens if we exchange the unitary groups for orthogonal or symplectic groups.