Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks

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


Introduction
Let G be a profinite group, and ϕ a (continuous) automorphism of G of finite order. We say for short that ϕ is a coprime automorphism of G if its order is coprime to the orders of elements of G (understood as Steinitz numbers), in other words, if G is an inverse limit of finite groups of order coprime to the order of ϕ. Coprime automorphisms of profinite groups have many properties similar to the properties of coprime automorphisms of finite groups. In particular, if ϕ is a coprime automorphism of G, then for any (closed) normal ϕ-invariant subgroup N the fixed points of the induced automorphism (which we denote by the same letter) in G{N are images of the fixed points in G, that is, C G{N pϕq " C G pϕqN{N. Therefore, if ϕ is a coprime automorphism of prime order p such that C G pϕq " 1, Thompson's theorem [15] implies that G is pronilpotent, and Higman's theorem [3] implies that G is nilpotent of class bounded in terms of p.
In our joint paper with Acciarri [2] we considered profinite groups admitting a coprime automorphism of prime order all of whose fixed points are right Engel elements. Recall that the n-Engel word ry, n xs is defined recursively by ry, 0 xs " y and ry, i`1 xs " rry, i xs, xs. An element g of a group G is said to be right Engel if for any x P G there is an integer n " npg, xq such that rg, n xs " 1. If all elements of a group are right Engel (therefore also left Engel), then the group is called an Engel group. By a theorem of Wilson and Zelmanov [20] based on Zelmanov's results [21,22,23] on Engel Lie algebras, an Engel profinite group is locally nilpotent. Recall that a group is said to be locally nilpotent if every finite subset generates a nilpotent subgroup. The following theorem was proved in [2]. Theorem 1.1 ( [2]). Suppose that ϕ is a coprime automorphism of prime order of a profinite group G. If every element of C G pϕq is a right Engel element of G, then G is locally nilpotent.
In this paper we consider profinite groups admitting a coprime automorphism of prime order all of whose fixed points have finite Engel sinks. Recall that Engel sinks are used to study generalizations of Engel conditions and are defined as follows.

Definition.
A left Engel sink of an element g of a group G is a set E pgq such that for every x P G all sufficiently long commutators rx, g, g, . . . , gs belong to E pgq, that is, for every x P G there is a positive integer lpx, gq such that rx, l gs P E pgq for all l ě lpx, gq.
(Thus, g is a left Engel element precisely when we can choose E pgq " t1u, and G is an Engel group when we can choose E pgq " t1u for all g P G.) Definition. A right Engel sink of an element g of a group G is a set Rpgq such that for every x P G all sufficiently long commutators rg, x, x, . . . , xs belong to Rpgq, that is, for every x P G there is a positive integer rpx, gq such that rx, r gs P Rpgq for all r ě rpx, gq.
(Thus, g is a right Engel element precisely when we can choose Rpgq " t1u, and G is an Engel group when we can choose Rpgq " t1u for all g P G.) Our main result concerning right Engel sinks is as follows. Note that if all elements of a profinite or compact group have finite or even countable left or right Engel sinks, then the group has a finite subgroup with locally nilpotent quotient [8,10,9,11]. Examples show that such a stronger conclusion does not hold under the hypotheses of Theorem 1.2, which is in a sense best-possible.
One of the important tools in the proof of Theorem 1.2 is a strengthened version of Neumann's theorem about BF C-groups from the recent paper of Acciarri and Shumyatsky [1]. The proof also makes use of the quantitative version for finite groups that we proved earlier in [12]. In that paper [12] we also proved that if a finite group G has a coprime automorphism ϕ of prime order such that all fixed points of ϕ have left Engel sinks of cardinality at most m, then G has a metanilpotent subgroup of index bounded in terms of m (examples show that here "metanilpotent" cannot be replaced by "nilpotent"). We prove the following profinite analogue of this result. Theorem 1.3. Let G be a profinite group admitting a coprime automorphism ϕ of prime order p. If all fixed points of ϕ have finite left Engel sinks, then G has an open subgroup that is an extension of a pronilpotent group by a nilpotent group of class hppq, where hppq is Higman's function depending only on p.
There are examples showing that in the conclusion of Theorem 1.3 "pronilpotent-bynilpotent" cannot be replaced even by "pronilpotent", in contrast to the stronger virtual local nilpotency conclusion of Theorem 1.2 about right Engel sinks. Similarly, if all fixed points of ϕ are left Engel elements, then the group G is an extension of a pronilpotent group by a nilpotent group of class hppq, where hppq is Higman's function (Remark 4.2), but G does not have to have an open locally nilpotent subgroup, unlike for the right Engel condition in Theorem 1.1. Thus, the situation with Engel sinks for fixed points of an automorphism is markedly different from the aforementioned results with conditions on Engel sinks of all elements of a profinite or compact group, where the finiteness (or countability) of right or left Engel sinks resulted in the same conclusion that the group is finite-by-(locally nilpotent).
It is worth mentioning that if, under the hypotheses of Theorems 1.2 (or 1.3), there is m P N such that all right (respectively, left) Engel sinks of fixed points of ϕ have cardinality at most m, then the conclusions can be strengthened, with bounds for the index of a locally nilpotent (respectively, pronilpotent-by-nilpotent) subgroup (Remarks 3.6 and 4.3).
We present preliminary material on profinite groups and left and right Engel sinks in § 1. Theorems 1.2 and 1.3 about right and left Engel sinks are proved in § 3 and § 4, respectively.
In § 5 we present examples showing that in some respects Theorems 1.2 and 1.3 cannot be improved.

Preliminaries
In this section we recall some definitions and general properties related to profinite groups and Engel sinks.
Our notation and terminology for profinite groups is standard; see, for example, [13] and [19]. A subgroup (topologically) generated by a subset S is denoted by xSy. By a subgroup we always mean a closed subgroup, unless explicitly stated otherwise. Recall that centralizers are closed subgroups, while commutator subgroups rB, As " xrb, as | b P B, a P Ay are the closures of the corresponding abstract commutator subgroups.
For a group A acting by automorphisms on a group B we use the usual notation for commutators rb, as " b´1b a and commutator subgroups rB, As " xrb, as | b P B, a P Ay, as well as for centralizers C B pAq " tb P B | b a " b for all a P Au and C A pBq " ta P A | b a " b for all b P Bu. A section A{B of a group G is a quotient of a subgroup A ď G by a normal subgroup B of A. The centralizer of a section is C G pA{Bq " tg P G | rA, gs ď Bu. The definition and some properties of coprime automorphisms of profinite groups were already mentioned at the beginning of the Introduction in § 1.
Recall that a pro-p group is an inverse limit of finite p-groups, a pronilpotent group is an inverse limit of finite nilpotent groups, a prosoluble group is an inverse limit of finite soluble groups. We denote by πpkq the set of prime divisors of k, where k may be a positive integer or a Steinitz number, and by πpGq the set of prime divisors of the orders of elements of a (profinite) group G. Let σ be a set of primes. An element g of a group is a σ-element if πp|g|q Ď σ, and a group G is a σ-group if all of its elements are σ-elements. We denote by σ 1 the complement of σ in the set of all primes. When σ " tpu, we write p-element, p 1 -element, etc. Profinite groups have Sylow p-subgroups and satisfy analogues of the Sylow theorems. Prosoluble groups satisfy analogues of the theorems on Hall π-subgroups. We refer the reader to the corresponding chapters in [13,Ch. 2] and [19,Ch. 2].
We denote by γ 8 pGq " Ş i γ i pGq the intersection of the lower central series of a group G. A profinite group G is pronilpotent if and only if γ 8 pGq " 1, which is also equivalent to G being the Cartesian product of its Sylow subgroups. Every profinite group G has a maximal normal pronilpotent subgroup denoted by F pGq. This subgroup has the following characterization, similar to that of the Fitting subgroup of a finite group. Proof. The intersection in question is clearly a closed normal subgroup. In any finite quotient of G, the image of this intersection is nilpotent by the well-known characterization of the Fitting subgroup of a finite group [14, 5.2.9]. Hence this intersection is contained in F pGq. Conversely, any element of F pGq clearly belongs to the Fitting subgroup of any finite quotient of G and therefore centralizes every chief factor of it.
We can define a profinite analogue of the Fitting series by setting F 1 pGq " F pGq, and then by induction F k`1 pGq being the inverse image of F pG{F k pGqq. It is natural to say that a profinite group has pronilpotent length l if F l pGq " G and l is minimal with this property. We record a useful elementary lemma about the pronilpotent series.
If g is a p-element in F 2 pGqzF pGq, then g induces by conjugation a non-trivial automorphism of the Hall p 1 -subgroup of F pGq.
Proof. (a) Clearly, F pGq ď F pHq. We now prove the reverse inclusion. Any chief factor A{B of a finite quotient of G is a section of G{F 2 pGq, or of F 2 pGq{F pGq, or of F pGq. An element of F pHq centralizes A{B in the first case because H ď F 2 pGq, in the second case because F 2 {F pGq is pronilpotent, and in the third case because F pGq ď H. Hence F pHq ď F pGq by Lemma 2.1.
(b) By Lemma 2.1 the element g must act nontrivially on some chief factor of a finite quotient of G by an open normal subgroups. Since g P F 2 pGq, such a chief factor must be a section of F pGq, and since g is contained in a Sylow p-subgroup containing the Sylow psubgroup of F pGq, such a chief factor must be a section of the Hall p 1 -subgroup of F pGq.
If P is a pro-p group, the Frattini subgroup of P is ΦpP q " rP, P sP p . If α is a coprime automorphism of P , then α acts nontrivially on P {ΦpP q. The Frattini subgroup of a pronilpotent group is the Cartesian product of the Frattini subgroups of its Sylow p-subgroups. It follows from Lemmas 2.1 and 2.2 that F pG{ΦpF pGqqq " F pGq{ΦpF pGqq. (2.1) Proof. The subgroup C G pγ 8 pGqq is closed and has finite index, since G{C G pγ 8 pGqq faithfully acts by automorphisms on γ 8 pGq; hence C G pγ 8 pGqq is an open normal subgroup. Any chief factor A{B of a finite quotient of G by an open normal subgroup is either a section of G{γ 8 pGq, which is pronilpotent, or of γ 8 pGq. Hence any element of C G pγ 8 pGqq centralizes A{B and the result follows by Lemma 2.1.
We recall the well-known consequence of the Baire Category Theorem (see [5, Theorem 34]).
Theorem 2.4. If a profinite group is a countable union of closed subsets, then one of these subsets has non-empty interior.
We now recall some general properties of Engel sinks. Clearly, the intersection of two left Engel sinks of a given element g of a group G is again a left Engel sink of g, with the corresponding function lpx, gq being the maximum of the two functions. Therefore, if g has a finite left Engel sink, then g has a unique smallest left Engel sink, which has the following characterization.
. If an element g of a group G has a finite left Engel sink, then g has a smallest left Engel sink E pgq and for every s P E pgq there is an integer k ě 1 such that s " rs, k gs.
The intersection of two right Engel sinks of a given element g of a group G is again a right Engel sink of g, with the corresponding function rpx, gq being the maximum of the two functions. Therefore, if g has a finite right Engel sink, then g has a unique smallest right Engel sink, which is henceforth denoted by Rpgq. It has the following characterization.
Lemma 2.6 ([9, Lemma 2.2]). If an element g of a group G has a finite right Engel sink, then g has a smallest right Engel sink Rpgq and for every z P Rpgq there are integers n ě 1 and m ě 1 and an element x P G such that z " rg, n xs " rg, n`m xs.
(Here, the elements x and numbers m, n can be different for different z.) Furthermore, for metabelian groups we have the following.
Lemma 2.7 ([9, Lemma 2.5]). If G is a metabelian group, then a right Engel sink of the inverse g´1 of an element g P G is a left Engel sink of g.
Remark 2.8. If ϕ is an automorphism of finite order p of a profinite group G and H is an open normal subgroup of G, then Ş p´1 i"0 H ϕ i is a ϕ-invariant open normal subgroup. Thus, ϕ-invariant open normal subgroups of G form a base of neighbourhoods of 1 in the profinite topology. We freely use this property throughout the paper without special references. Remark 2.9. If every element of a subgroup C has a finite right Engel sink in a group G, then this condition is inherited by the image of C X A in every section A{B, and we shall use this property without special references. The same applies to a subgroup in which every element has a finite left Engel sink.

Right Engel sinks
In this section we prove Theorem 1.2 concerning right Engel sinks of fixed points of an automorphism.
Lemma 3.1. If G is a pronilpotent group and an element g P G has a finite right Engel sink, then in fact Rpgq " t1u, that is, g is a right Engel element.
Proof. Since Rpgq is finite, there is an open normal subgroup N such that Rpgq X N " t1u. If z P Rpgq, then by Lemma 2.6 there are integers n ě 1 and m ě 1 and x P G such that z " rg, n xs " rg, n`m xs. Therefore the image of z in G{N must be trivial, since G{N is nilpotent. Hence z P N X Rpgq " t1u.
Combining Lemma 3.1 with Theorem 1.1 we obtain the following.
Corollary 3.2. If a pronilpotent group G admits a coprime automorphism of prime order such that every fixed point has a finite right Engel sink, then G is locally nilpotent.
The following quantitative version of Theorem 1.2 for finite groups was proved in [12]. . Let G be a finite group admitting an automorphism ϕ of prime order coprime to |G|. Let m be a positive integer such that every element g P C G pϕq has a right Engel sink Rpgq of cardinality at most m. Then G has a nilpotent normal subgroup of m-bounded index.
In the proof of Theorem 1.2 we will combine this result with Corollary 3.2 and a reduction to the case of uniformly bounded sizes of right Engel sinks of fixed points.
Proof of Theorem 1.2. Recall that G is a profinite group admitting a coprime automorphism ϕ of prime order such all fixed points of ϕ have finite right Engel sinks; we need to produce an open locally nilpotent subgroup. By Corollary 3.2 any ϕ-invariant pronilpotent subgroup of G is locally nilpotent and therefore it is sufficient to produce an open pronilpotent subgroup.
Let g P C G pϕq and let N g be an open normal subgroup such that N g X Rpgq " t1u. Then g is a right Engel element of the subgroup N g xgy. By Baer's theorem [14, 12.3.7], in every finite quotient of N g xgy the image of g belongs to the hypercentre. Therefore the subgroup rN g , gs is pronilpotent.
Let r N g be the normal closure of rN g , gs in G. Since rN g , gs is normal in the subgroup N g , which has finite index, rN g , gs has only finitely many conjugates. Hence r N g is a product of finitely many normal subgroups of N g , each of which is pronilpotent, and therefore r N g is pronilpotent. Therefore all the subgroups r N g are contained in the largest normal pronilpotent subgroup F pGq.
The imageḡ of every element g P C G pϕq in s G " G{F pGq has finite conjugacy classḡ G , since rg, N g s ď F pGq and N g has finite index in G. We now use a strengthened version of Neumann's theorem about BF C-groups and a lemma about finite conjugacy classes in profinite groups from the recent paper of Acciarri and Shumyatsky [1]. Namely, by [1,Lemma 4.2] there is an integer n such that |ḡ G | ď n for everyḡ P C s G pϕq. Let H " xC s G pϕq G y be the abstract normal closure of C s G pϕq in s G. Then by [1, Theorem 1.1] the derived subgroup H 1 is finite (of n-bounded order). In particular, H 1 is a closed subgroup of s G. Let r H be the topological closure of H in s G. Since H{H 1 is abelian, r H{H 1 is also abelian. (We had to consider the abstract normal closure first, since [1, Theorem 1.1] is stated for abstract groups; but it is clear that it also works for profinite groups as shown above.) Note that C s G pϕq ď r H and therefore s G{ r H is nilpotent by the theorems of Thompson [15] and Higman [3]. Let N be a ϕ-invariant open normal subgroup of G containing F pGq such that s N X H 1 " 1. Then N{F pNq is abelian-by-nilpotent. Replacing G with N we can assume from the outset that G{F pGq is soluble and proceed by induction on the derived length of it.
The main case is when G{F pGq is abelian. Indeed, in the general case, by induction hypothesis, Then N{F pNq is abelian, so we may assume that G{F pGq is abelian from the outset. We need to show that G{F pGq is finite.
We write F " F pGq to lighten the notation. Since F pG{ΦpF qq " F {ΦpF q by (2.1), we can assume that ΦpF q " 1. In particular, then G is metabelian. Proof. By Lemma 2.7 we have E pgq Ď Rpg´1q in a metabelian group. Hence all elements of C G pϕq have finite left Engel sinks. Every subset E k " tx P C G pϕq | |E pxq| ď ku is closed in the induced topology of C G pϕq. Indeed, this is equivalent to the complement of C G pϕqzE k being an open subset of C G pϕq. For every element g P C G pϕqzE k we have |E pgq| ě k`1, so there are distinct elements z 1 , z 2 , . . . , z k`1 P E pgq. By Lemma 2.5 we can write for every i " 1, . . . , k`1 z i " rz i , g, . . . , gs, where g is repeated k i ě 1 times. (3.1) Let N be an open normal subgroup of G such that the images of z 1 , z 2 , . . . , z k`1 are distinct elements in G{N. Then equations (3.1) show that for any u P N X C G pϕq the Engel sink E pguq contains an element in each of the k`1 cosets z i N. This means that the whole coset gpN X C G pϕqq is contained in C G pϕqzE k . Thus every element of C G pϕqzE k has a neighbourhood contained in C G pϕqzE k , which is therefore an open subset of C G pϕq.
Since C G pϕq " Ť k E k , by the Baire Category Theorem 2.4 there is m P N, an open (in the induced topology) normal subgroup C 1 , and a coset c 0 C 1 such that |E pc 0 xq| ď m for any x P C 1 . We now obtain that |E pxq| is m-bounded for any x P C 1 . Indeed, by [9,Lemma 2.5], in a metabelian group, if E pgq is finite, then E pgq is a normal subgroup. In the quotient Ď M " M{`E pc 0 qE pc 0 xq˘, both Ď M 1 xc 0 y and Ď M 1 xc 0x y are normal locally nilpotent subgroups. Hence their product, which containsx, is also a locally nilpotent subgroup by the Hirsch -Plotkin theorem [14, 12.1.2]. As a result, E pxq ď E pc 0 qE pc 0 xq and therefore |E pxq| ď |E pc 0 q|¨|E pc 0 xq| ď m 2 for any x P C 1 .
To prove that C G pϕqF {F is finite, it remains to show that C 1 F {F is finite. For that we use the following lemma.

Lemma 3.5 ([12, Lemma 2.4]).
Suppose that V is an abelian finite group, and U a group of coprime automorphisms of V . If |rV, us| ď n for every u P U , then |rV, Us| is n-bounded, and therefore |U| is also n-bounded.
Let q be any prime in πpC 1 F {F q, and let Q be a Sylow q-subgroup of C 1 F . Let f pnq be the function furnished by Lemma 3.5 as a bound in terms of n for |U|. We claim that |QF {F | ď f pm 2 q; this will imply that C 1 F {F is finite. Note that every element of QzF acts non-trivially on the Hall q 1 -subgroup F q 1 of F . For every u P Q, the left Engel sink E puq is a normal subgroup of order ď m 2 by (3.2). Since F q 1 xuy{E puq is pronilpotent and u induces a coprime automorphism on F q 1 , we have F q 1 " E puqC F q 1 puq. Hence rF q 1 , us " rE puqC F q 1 puq, us " rE puq, us " E puq, where the last equality holds by the minimality of E puq. Therefore in every finite quotient s G of G by a ϕ-invariant open normal subgroup we have | s Q{C s Q p s F q 1 q| ď f pm 2 q by Lemma 3.5. Hence |QF {F | " |Q{C Q pF q 1 q| ď f pm 2 q, as claimed.
Since C 1 has finite index in C G pϕq and C 1 F {F is finite, we conclude that C G pϕqF {F is finite.
Let N be a ϕ-invariant open normal subgroup of G containing F such that N XC G pϕq ď F . Then Since F pNq " F , we have C N pϕq ď F pNq. Replacing G with N we can assume from the outset that C G pϕq ď F . Every subset R k " tx P C G pϕq | |Rpxq| ď ku is closed in the induced topology of C G pϕq. Indeed, this is equivalent to the complement of R k being an open subset of C G pϕq. For every element g P C G pϕqzR k we have |Rpgq| ě k`1 and there are distinct elements z 1 , z 2 , . . . , z k`1 P Rpgq. Using Lemma 2.6 we can write for every i " 1, . . . , k`1 z i " rg, n i x i s " rg, n i`mi xs for some x i P G, n i ě 1, m i ě 1. (3.3) Let N be an open normal subgroup of G such that the images of z 1 , z 2 , . . . , z k`1 are distinct elements in G{N. Then equations (3.3) show that for any u P N XC G pϕq the right Engel sink Rpguq contains an element in each of the k`1 cosets z i N. Thus, the coset gpN X C G pϕqq is contained in C G pϕqzR k . This means that every element of C G pϕqzR k has a neighbourhood contained in C G pϕqzR k , which is therefore an open subset of C G pϕq.
Since C G pϕq " Ť k R k , by the Baire Category Theorem 2.4 there is m P N, an open (in the induced topology) normal subgroup C 1 of C G pϕq, and a coset c 0 C 1 such that |Rpc 0 xq| ď m for any x P C 1 . (3.4) By the standard commutator formula rxy, zs " rx, zs y ry, zs, using the fact that F is abelian we have rab, x, . . . , x looomooon n s " ra, x, . . . , x looomooon n s¨rb, x, . . . , x looomooon n s for any a, b P F , any x P G, and any n P N. Therefore we obtain that Rpabq Ď RpaqRpbq for any a, b P C G pϕq ď F . The same commutator formula shows that Rpa´1q " tz´1 | z P Rpaqu, so that |Rpa´1q| " |Rpaq|. From (3.4) we now obtain |Rpxq| ď |Rpc´1 0 q|¨|Rpc 0 xq| ď m 2 for any x P C 1 . Remark 3.6. If, under the hypotheses of Theorem 1.2 there is a positive integer n such that all fixed points of ϕ have finite right Engel sinks of cardinality at most n, then the group G has a locally nilpotent subgroup of finite n-bounded index. This immediately follows from Theorem 3.3 applied to finite quotients of G by a ϕ-invariant open normal subgroup: every such quotient has a nilpotent subgroup of index at most f pnq for a function f pnq depending only on n. Therefore G has a ϕ-invariant open pronilpotent subgroup M of index at most f pnq, which is locally nilpotent by Corollary 3.2.

Left Engel sinks
In this section we prove Theorem 1.3 concerning left Engel sinks of fixed points. We begin with the following lemma.
Lemma 4.1. Suppose that G " F 2 pGq is a profinite group of pronilpotent length 2 admitting a coprime automorphism ϕ of prime order such that all elements of C G pϕq have finite left Engel sinks.
(a) Then C G{F pGq pϕq is finite.
(c) If all elements of C G pϕq are left Engel elements, then C G pϕq ď F pGq.
Proof. We write F " F pGq to lighten the notation. Let ΦpF q be the Frattini subgroup of F . Since F pG{ΦpF qq " F {ΦpF q by (2.1), we can assume that F is abelian in all parts of the lemma.
(a) Since the group F C G pϕq{F is pronilpotent and all its elements have finite left Engel sinks, this group is locally nilpotent by [8,Lemma 4.2]. We further claim that all elements of F C G pϕq have finite left Engel sinks in F C G pϕq. Indeed, let g " uc and h " vd, where u, v P F and c, d P C G pϕq. For some k the commutator rh, k gs belongs to F , since xu, vyF {F is nilpotent. Then rh, k`n gs " rrh, k gs, n cs for any n, since F is abelian. As a result, E pgq is contained in E pcq, which is finite by hypothesis.
Applying [8,Theorem 1.2] we obtain that γ 8 pF C G pϕqq is finite. By Lemma 2.3, C G pγ 8 pF C G pϕqqq is a closed normal pronilpotent subgroup, which has finite index in F C G pϕq. It follows that F pF C G pϕqq has finite index in F C G pϕq, and the result follows, since F pF C G pϕqq " F by Lemma 2.2.
(b) If there is m P N such that |E pcq| ď m for all c P C G pϕq, then the above argument shows that |E pgq| ď m for all g P F C G pϕq. By [8, Theorem 3.1] then |γ 8 p s F C s G pϕqq| is m-bounded for every finite quotient s G of G by a ϕ-invariant open normal subgroup. Hence |γ 8 pF C G pϕqq| is also m-bounded, and so is the index of F in F C G pϕq by the above argument.
(c) Since all elements of C G pϕq are left Engel elements, the above argument shows that C G pϕqF is an Engel group, and therefore pronilpotent. Since F " F pF C G pϕqq by Lemma 2.2, we obtain F " F C G pϕq.
Proof of Theorem 1.3. Recall that G is a profinite group admitting a coprime automorphism ϕ of prime order such that all elements of C G pϕq have finite left Engel sinks. We need to show that G has an open pronilpotent-by-nilpotent subgroup. First we perform reduction to the case of pronilpotent length 3, that is, G " F 3 pGq. Since all elements of C G pϕq have finite left Engel sinks, by [8,Theorem 1.2] there is a finite normal subgroup C 0 of C G pϕq such that C G pϕq{C 0 is locally nilpotent. Let N be a ϕ-invariant open normal subgroup of G such that N X C 0 " 1. Then C N pϕq is locally nilpotent. Then the centralizer C s N pϕq is nilpotent for every finite quotient s N of N by a ϕ-invariant open normal subgroup. Wang and Chen [18] used the classification of finite simple groups to prove that a finite group admitting a coprime automorphism of prime with nilpotent fixed-point subgroup is soluble. Furthermore, by a theorem of Turull [17] (the best-possible improvement of the earlier result of Thompson [16]), the Fitting height of s N is at most 3. Thus, every finite quotient of N by an open normal subgroup has Fitting height at most 3; hence N has pronilpotent length at most 3. Replacing G with N, we can assume from the outset that G " F 3 pGq.
By Lemma 4.1, both C G{F 2 pGq pϕq and C F 2 pGq{F pGq pϕq are finite. Hence, C G{F pGq pϕq is finite. Let H be a ϕ-invariant open normal subgroup containing F pGq such that HXC G pϕq ď F pGq. Then C H pϕq ď F pGq " F pHq. By the theorems of Thompson [15] and Higman [3] the quotient H{F pHq is nilpotent of class at most hppq, where p " |ϕ|.
Thus, G has an open subgroup that is an extension of a pronilpotent group by a nilpotent group of class at most hppq.

Remark 4.2.
If, under the hypotheses of Theorem 1.3 all fixed points of ϕ are left Engel elements, then the group G is an extension of a pronilpotent group by a nilpotent group of class hppq, where hppq is Higman's function. Indeed, then C G pϕq ď F pGq by Lemma 4.1(c). Then G{F pGq is nilpotent of class at most hppq, where p " |ϕ|, by the theorems of Thompson [15] and Higman [3]. Gq in s G is m-bounded. Hence the index of F 2 pGq in G is also m-bounded. Furthermore, by Lemma 4.1(b) the order |CḠ {F pḠq pϕq| is m-bounded. By Khukhro's theorem [6,Theorem 2], thenḠ{F pḠq has a subgroup of pp, mq-bounded index that is nilpotent of p-bounded class gppq. This implies that s G has an open subgroup of pp, mq-bounded index that is nilpotent of class gppq.

Examples
Here we present examples showing that in some respects Theorems 1.2 and 1.3 cannot be improved.
Example 5.1. Let V be an elementary abelian group of order 7 2 , and D 6 " xay¸xby a group of automorphisms of V such that a 3 " 1, b 2 " 1, C V paq " 1, and |C V pbq| " 7. Let F " ś 8 i"1 V i be the Cartesian product of isomorphic copies V i of V as F 7 D 6 -modules. Then D 6 can be regarded as a group of automorphisms of F . Let G " F xay and let ϕ be the automorphism of G of order 2 induced by b. Then C G pϕq " ś 8 i"1 C V i pϕq. Using the fact that all the V i are isomorphic F 7 xay-modules, one can show that for any c P C G pϕq the right Engel sink Rpcq is finite and, moreover, the sizes of these sinks are uniformly bounded. At the same time, γ 8 pGq " F is infinite. This example shows that under the hypotheses of Theorem 1.2 one cannot obtain a finite subgroup with a pronilpotent quotient.
Example 5.2. For the same V and D 6 " xay¸xby as in Example 5.1, let F " ś n i"1 V i be a finite direct product of n copies V i of V as F 7 D 6 -modules. Let G " F xay and let ϕ be the automorphism of G of order 2 induced by b. Then C G pϕq " ś n i"1 C V i pϕq. There is a constant m independent of n such that |Rpcq| ď m for any c P C G pϕq. In these examples, γ 8 pGq " F , so |γ 8 pGq| cannot be bounded in terms of m (and |ϕ|). This shows that the conclusion of Theorem 3.3 ([12, Theorem 1.4]) also cannot be improved in this respect. Example 5.3. For the same V and D 6 " xay¸xby as in Example 5.1, let H " ś 8 i"1 V ip xa i y¸xb i yq be the Cartesian product of isomorphic copies V i¸p xa i y¸xb i yq of the semidirect product V¸pxay¸xbyq with V i , a i , b i naturally corresponding to V, a, b. Let G " ś 8 i"1 V i¸x a i y and let ϕ be the automorphism of G of order 2 induced by the 'diagonal' ś 8 i"1 b i . Then F pGq " ś 8 i"1 V i and C G pϕq " ś 8 i"1 C V i pϕq. Since F is abelian, all left Engel sinks of fixed points of ϕ are trivial. At the same time, G{F pGq is infinite. This example shows that under the hypotheses of Theorem 1.3 one cannot obtain an open pronilpotent subgroup (and the more so, a finite normal subgroup with a pronilpotent quotient).
Example 5.4. Similarly to Example 5.2, using finite direct products H " ś n i"1 V i¸p xa i yx b i yq instead of the Cartesian product, we obtain examples of finite groups G with a coprime automorphism ϕ of order 2 such that all elements of C G pϕq have trivial left Engel sinks. These examples show that the conclusion of [12, Theorem 1.3] giving a bound for the index of F 2 pGq cannot be improved to a bound for the index of F pGq.
Example 5.5. Let p be an odd prime and let A " xa 1 yˆxa 2 y » Z pˆZp be a direct product of two copies of p-adic integers regarded as procyclic pro-p groups with (topological) generators a 1 , a 2 . Let B " xby » Z p be another procyclic pro-p group with generator b. We define an action of b by an automorphism on xa 1 y by setting a b " a p`1 . Then we define the action of b by an automorphism on xa 2 y as the inverse of the automorphism a 2 Þ Ñ a p`1 2 . The resulting semidirect product G " A¸B admits an automorphism ϕ of order 2 such that b ϕ " b´1 and a ϕ 1 " a 2 . Then C G pϕq ď A and therefore all left Engel sinks of fixed points of ϕ are trivial. This example shows that a pronilpotent group with a coprime automorphism of prime order all of whose fixed points have trivial left Engel sinks does not have to have an open locally nilpotent subgroup, in contrast to Theorem 1.1 concerning right Engel sinks.