A restriction on centralizers in finite groups

Abstract

Abstract. For a given m ≥ 1, we consider the finite non-abelian groups G for which |CG(g) : g | ≤ m for every 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 = p k to be a power of p, we show that |G| ≤ p 2k+2 with the only exception of Q8. 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.

A finiteness condition on centralizers in locally nilpotent groups

“…In this paper we continue our study, introduced in [3], of groups with a finiteness condition on centralizers of elements, which was in turn inspired by the results in [1,2] and [11,12]. Following [3], we say that a group G is an FCI-group provided that (1) |C G (x) : x | < ∞ for every x ⋪ G, and if there exists n such that (2) |C G (x) : x | ≤ n for every x ⋪ G, then we say that G is a BCI-group. It is easy to see that free groups satisfy the FCI-condition for every non-trivial element, and it is also known that the same happens with hyperbolic torsion-free groups [4].…”

Some finiteness conditions on normalizers or centralizers in groups

“…Motivated by results of Q. Zhang and Gao [8] and X. Zhang and Guo [9], we considered in [2] the finite p-groups G, where p is a prime, satisfying one of the following three conditions for a given n: that |N G (H) : H| ≤ n for every non-normal subgroup H of G, or that either |N G ( x ) : x | ≤ n or |C G (x) : x | ≤ n for every non-normal cyclic subgroup x of G. In [1] we also dealt with the last of these conditions with non-central elements instead of elements generating a non-normal subgroup, in the case of general finite groups.…”

A finiteness condition on centralizers in locally finite groups

“…In the present paper we focus on a natural finiteness condition on centralizers of elements in a locally finite group. Our motivation stems from our previous works [1] and [2], where we dealt with two restrictions on centralizers in finite p-groups, namely that |C G (x) : x | ≤ n either for all x ∈ G Z(G), or for all x ∈ G such that x ⋪ G. Actually, in [1], we studied the first of these conditions in the case of general finite groups.…”

“…Given a group G, we say that G is an FCI-group (FCI for 'finite centralizer index') provided that (1) |C G (x) : x | < ∞ for every x ⋪ G.…”

“…For periodic groups this condition is the same as the simpler |C G (x)| < ∞ for every x ⋪ G. We can then ask for the existence of a uniform bound for the indices in (1), thus getting the condition that, for some positive integer n, (2) |C G (x) : x | ≤ n for every x ⋪ G, in which case we speak of BCI-group (BCI for 'bounded centralizer index'). It is also natural to look at the same condition as in (1) with non-central elements instead of elements generating a non-normal subgroup, which reads However, this latter condition has such a strong implication in our context that it does not make sense to consider it. In fact, it is easy to see that every periodic group containing an infinite abelian subgroup, and in particular every infinite locally finite group, is abelian whenever (3) is satisfied.…”