A restriction on centralizers in finite groups

For a given m >= 1, we consider the finite non-abelian groups G for which vertical bar C-G(g) : < g >vertical bar <= m for every g epsilon 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 vertical bar G vertical bar <= 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. (C) 2013 Elsevier Inc. All rights reserved.