Profinite groups with restricted centralizers of powers
Cristina Acciarri, Pavel Shumyatsky

TL;DR
This paper investigates profinite groups with a restriction on centralizers of powers, showing such groups have an open normal subgroup with a quotient of finite exponent.
Contribution
It introduces a new class of profinite groups with restricted centralizers of powers and proves they have an open normal subgroup with a finite exponent quotient.
Findings
Groups with restricted centralizers of powers have an open normal subgroup T.
The quotient G/Z(T) has finite exponent.
This generalizes previous results on groups with restricted centralizers.
Abstract
A group is said to have restricted centralizers if for every the centralizer either is finite or has finite index in . Shalev showed that a profinite group with restricted centralizers is virtually abelian. Here we take interest in profinite groups for which there is an integer such that is either finite or open whenever . It is shown that such a group has an open normal subgroup with the property that has finite exponent.
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