A restriction on centralizers in finite groups

TL;DR

This paper investigates finite non-abelian groups with restrictions on the size of centralizers relative to elements outside the center, establishing bounds on their order based on a parameter m and prime divisors.

Contribution

It provides bounds on the order of such groups, especially for p-groups, and introduces a method to relate centralizer restrictions to group order.

Findings

Bound |G| in terms of m and largest prime divisor

For p-groups, |G|<=p^{2k+2} with exception of Q_8

Order bounded by a function of m for nilpotent groups

Abstract

For a given m>=1, we consider the finite non-abelian groups G for which |C_G(g):<g>|<=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|<=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.