On page 99 of "Structure of linear groups" by John D Dixon there's an exercise right after
Theorem: Let $G$ be a finite non-modular linear group of degree $n$. Then $G$ has a normal abelian subgroup $A$ with index $[G:A]\leq (49n)^{n^2}$.
Exercise 1: Let $G$ be a finite non-modular linear group of degree $n$. Show that if $G$ has an abelian subgroup $B\supseteq Z(G)$ with $[B:Z(G)]>(4\pi)^n$, then $G$ has a normal abelian subgroup $A$ with $B\supseteq A\supsetneq Z(G)$. [Hint: We may suppose $G$ consists of unitary matrices, and the elements of $B$ are diagonal.]
I have absolutely no idea what to do here. Neither how to apply the theorem nor how to get to $(4\pi)^n$. I don't know how to make any use of the hint either. Any help would be appreciated.