Profinite Groups with Restricted Centralizers

Abstract

Abstract. Let G be a profinite group in which every centralizer Cq{x) (x 6 G) is either finite or of finite index. It is shown that G is finite-by-abelian-byfinite. Moreover, if, in addition, G is a just-infinite pro-p group, then it has the structure of a p-adic space group whose point group is cyclic or generalized quaternion.

Profinite groups with restricted centralizers of $π$-elements

“…Thus, the improvement over Theorem 1.1 is twofold -the result now covers the case of π-elements and provides additional details clarifying the structure of groups in question. Furthermore, it is easy to see that Theorem 1.3 extends the Shalev result [13] which can be recovered by considering the case where π = π(G) is the set of all prime divisors of the order of G.…”

“…Hence L is prosoluble. Let K be a Hall π-subgroup of L. Since any element in K has restricted centralizer, Shalev's result [13] shows that K is virtually abelian. We therefore can choose an open normal subgroup J in L such that J ∩ K is abelian.…”

“…A group G is said to have restricted centralizers if for each g in G the centralizer C G (g) either is finite or has finite index in G. This notion was introduced by Shalev in [13] where he showed that a profinite group with restricted centralizers is virtually abelian. We say that a profinite group has a property virtually if it has an open subgroup with that property.…”

Finite-by-nilpotent groups and a variation of the BFC-theorem

“…For example a direct product of infinitely many non-abelian finite groups is FC but not BFC. On the other hand, Shalev observed in [13] that a profinite FC-group is necessarily BFC. Here we will prove the following related result.…”

A stronger form of Neumann's BFC-theorem

“…In Section 4 we handle profinite groups with restricted centralizers. A group G is said to have restricted centralizers if for each g in G the centralizer C G (g) either is finite or has finite index in G. This notion was introduced by Shalev in [15] where he showed that a profinite group with restricted centralizers is virtually abelian. We say that a profinite group has a property virtually if it has an open subgroup with that property.…”