A restriction on centralizers in finite groups

For a given m⩾1m⩾1, we consider the finite non-abelian groups G   for which |CG(g):〈g〉|⩽m|CG(g):〈g〉|⩽m for every g∈G∖Z(G)g∈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=pkm=pk to be a power of p  , we show that |G|⩽p2k+2|G|⩽p2k+2 with the only exception of Q8Q8. 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.