A restriction on centralizers in finite groups

For a given m>0, we consider the finite non-abelian groups G for which |C_G(g) : <g>| is less than or equal to m for every g in G \ Z(G). We show that the order of G can be bounded in terms of m and the largest prime divisor of the order of G. Our approach relies on dealing first with the case where G is a non-abelian finite p-group. In that situation,if we take m=p^k to be a power of p, we show that |G| is less than or equal to p^(2k+2) with the only exception of Q_8. This bound is best possible, and implies that the order of G can be bounded by a function of m alone in the case of nilpotent groups.