On profinite groups with automorphisms whose fixed points have countable Engel sinks

Abstract: An Engel sink of an element

g

of a group

G

is a set

m a t h s c r E (g)

such that for every

x i n G

all sufficiently long commutators

[. . . [[x, g], g], d o t s, g]

belong to

m a t h s c r E (g)

. (Thus,

g

is an Engel element precisely when we can choose

m a t h s c r E (g) = 1

.) It is proved that if a profinite group

G

admits an elementary abelian group of automorphisms

A

of coprime order

q^{2}

for a prime

q

such that for each

a i n A s e t m i n u s 1

every element of the centralizer

C_{G} (a)

has a countable (or finite) Engel sink, then

G

has a finite normal subgroup

N

such that

G / N

is locally nilpotent.

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