A finiteness condition on centralizers in locally nilpotent groups

Abstract

We give a detailed description of infinite locally nilpotent groups G such that the index |C G (x) : x | is finite, for every x G. We are also able to extend our analysis to all non-periodic groups satisfying a variation of our condition, where the requirement of finiteness is replaced with a bound.

Some finiteness conditions on normalizers or centralizers in groups

“…Since x 2 = g 2 , the order of x is either 2 or 4. Thus As already indicated in the proof of the previous theorem, according to [4,Theorem 4.3], the structure of non-periodic BCI-groups is as described in (i) above, but only requiring that the 2-rank of A should be finite. Hence we need to impose the extra condition that also the 0-rank of A should be finite for a non-periodic BCI-group to be a BNI-group.…”

“…(i) Let us first assume that G is a BNI-group. Then G is also a BCIgroup, and then, by Theorem 4.3 of [4], G is either abelian or of the form G = g, A , where A is a non-periodic abelian group of finite 2-rank, and g is an element of order at most 4 such that g 2 ∈ A and a g = a −1 for all a ∈ A. Thus we only need to prove that, in the latter case, A is also of finite 0-rank.…”

“…By Theorem 2.4, we only need to deal with non-periodic groups. On the other hand, we know from [4,Corollary 4.4] that in this situation BCI-groups are abelian, and so the same holds for BNI-groups. Hence it suffices to study the case of FNI-groups.…”

“…Hence it suffices to study the case of FNI-groups. To this purpose, we recall the characterization of non-periodic locally nilpotent FCI-groups which we obtained in [4,Theorem 3.7].…”

“…Later, in [3] and [4], we extended our earlier work to the realm of infinite groups. Following [3], we say that a group G is an FCI-group (FCI for 'finite centralizer index') provided that (1) |C G (x) : x | < ∞ for every x ⋪ G, and if there exists n such that…”