AbstractA right Engel sink of an element g of a group G is a set $${{\mathscr {R}}}(g)$$
R
(
g
)
such that for every $$x\in G$$
x
∈
G
all sufficiently long commutators $$[...[[g,x],x],\dots ,x]$$
[
.
.
.
[
[
g
,
x
]
,
x
]
,
⋯
,
x
]
belong to $${\mathscr {R}}(g)$$
R
(
g
)
. (Thus, g is a right Engel element precisely when we can choose $${{\mathscr {R}}}(g)=\{ 1\}$$
R
(
g
)
=
{
1
}
.) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$
φ
of prime order such that every fixed point of $$\varphi $$
φ
has a finite right Engel sink, then G has an open locally nilpotent subgroup. A left Engel sink of an element g of a group G is a set $${{\mathscr {E}}}(g)$$
E
(
g
)
such that for every $$x\in G$$
x
∈
G
all sufficiently long commutators $$[...[[x,g],g],\dots ,g]$$
[
.
.
.
[
[
x
,
g
]
,
g
]
,
⋯
,
g
]
belong to $${{\mathscr {E}}}(g)$$
E
(
g
)
. (Thus, g is a left Engel element precisely when we can choose $${\mathscr {E}}(g)=\{ 1\}$$
E
(
g
)
=
{
1
}
.) We prove that if a profinite group G admits a coprime automorphism $$\varphi $$
φ
of prime order such that every fixed point of $$\varphi $$
φ
has a finite left Engel sink, then G has an open pronilpotent-by-nilpotent subgroup.
On profinite groups with automorphisms whose fixed points have countable Engel sinks