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

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 finite group $G$ admits an automorphism $\varphi $ of prime order coprime to $|G|$ such that for some positive integer $m$ every element of the centralizer $C_G(\varphi )$ has a left Engel sink of cardinality at most $m$, then the index of the second Fitting subgroup $F_2(G)$ is bounded in terms of $m$. 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 finite group $G$ admits an automorphism $\varphi$ of prime order coprime to $|G|$ such that for some positive integer $m$ every element of the centralizer $C_G(\varphi )$ has a right Engel sink of cardinality at most $m$, then the index of the Fitting subgroup $F_1(G)$ is bounded in terms of $m$.


Introduction
We use the left-normed simple commutator notation  ] where b is repeated k times. Recall that an element g of a group G is called a left Engel element if for every x ∈ G the equation [x, n g] = 1 holds for some n = n(x) depending on x. An element g ∈ G is called a right Engel element if for every x ∈ G the equation [g, n x] = 1 holds for some n = n(x). An Engel group is defined as a group in which every element is a left Engel element (equivalently, every element is a right Engel element). Finite Engel groups are nilpotent by Zorn's theorem [21, 12.3.4]. Moreover, by Baer's theorem [21, 12.3.7], a left Engel element of a finite group belongs to its Fitting subgroup, and a right Engel element of a finite group belongs to its hypercentre. For many classes of groups theorems related to properties of Engel elements have been proved, achieving various structural properties related to nilpotency. For example, Wilson and Zelmanov [25] proved that profinite Engel groups are locally nilpotent. At the same time, Golod's examples [6] show that Engel groups in general may not be locally nilpotent.

Definition.
A left Engel sink of an element g of a group G is a set E (g) such that for every x ∈ G all sufficiently long commutators [x, g, g, . . . , g] belong to E (g), that is, for every x ∈ G there is a positive integer l(x, g) such that [x, l g] ∈ E (g) for all l l(x, g).
(Thus, g is a left Engel element precisely when we can choose E (g) = {1}, and G is an Engel group when we can choose E (g) = {1} for all g ∈ G.) Definition. A right Engel sink of an element g of a group G is a set R(g) such that for every x ∈ G all sufficiently long commutators [g, x, x, . . . , x] belong to R(g), that is, for every x ∈ G there is a positive integer r(x, g) such that [x, r g] ∈ R(g) for all r r(x, g).
(Thus, g is a right Engel element precisely when we can choose R(g) = {1}, and G is an Engel group when we can choose R(g) = {1} for all g ∈ G.) In [14,17] we considered finite, profinite, and compact (Hausdorff) groups in which every element has a finite or countable left Engel sink and proved that then the group is close to being locally nilpotent in a certain precise sense. In [16,18] we obtained similar results concerning right Engel sinks.
When G is a finite group, then every element has the smallest left Engel sink, since the intersection of two left Engel sinks E ′ (g) and E ′′ (g) is again a left Engel sink of g. Similarly, every element g ∈ G has the smallest right Engel sink. In this paper we shall always use the notation E (g) and R(g) to denote the smallest left and right Engel sinks of g, respectively, thus eliminating the ambiguity of this notation in the above definitions.
For finite groups we proved in [14] and [16] the following quantitative results. . Let G be a finite group, and m a positive integer. Suppose that every element g ∈ G has a right Engel sink R(g) of cardinality at most m. Then G has a normal subgroup N of order bounded in terms of m such that G/N is nilpotent.
As is well known, often strong consequences for the structure of the group can be derived from certain conditions imposed on fixed points of automorphisms. Some of the best-known examples include Thompson's theorem [23] on the nilpotency of a finite group with a fixedpoint-free automorphism of prime order, as well as numerous other papers on finite groups admitting automorphisms with various restrictions on their fixed points. This type of results in relation to automorphisms whose fixed points have restrictions on their Engel sinks were recently obtained in [1,2,3,4,5,20].
The purpose of this paper is to obtain the following 'automorphism extensions' of Theorems 1.1 and 1.2; one may also regard these extensions as results on 'almost fixed-point-free' automorphisms. Theorem 1.3. 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 ∈ C G (ϕ) has a left Engel sink E (g) of cardinality at most m. Then G has a metanilpotent normal subgroup of index bounded in terms of m only.
The conclusion for right Engel sinks is stronger. Note that while it is well-known that the inverse of a right Engel element is left Engel, there is no such a straightforward connection between left and right Engel sinks, and Theorem 1.4 is not a consequence of Theorem 1.3. Theorem 1.4. 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 ∈ C G (ϕ) has a right Engel sink R(g) of cardinality at most m. Then G has a nilpotent normal subgroup of index bounded in terms of m only.
Recall that the Fitting series starts with the Fitting subgroup F 1 (G) = F (G), and by induction, F k+1 (G) is the inverse image of F (G/F k (G)). Throughout the paper we shall write, say, "(a, b, . . . )-bounded" to abbreviate "bounded above in terms of a, b, . . . only". Thus, the conclusion of Theorem 1.3 states that the index of F 2 (G) is m-bounded, and the conclusion of Theorem 1.4 states that the index of F (G) is m-bounded.
The results of Theorems 1.3 and 1.4 are in a sense best-possible. There are easy examples showing that in Theorem 1.3 one cannot obtain a bound in terms of m (or even in terms of |ϕ| and m) for the index of F (G), nor can one prove that there is a normal subgroup of (|ϕ|, m)-bounded order with metanilpotent quotient. In Theorem 1.3 one cannot prove that there is a normal subgroup of (|ϕ|, m)-bounded order with nilpotent quotient.
We do not know whether the results of Theorems 1.3 and 1.4 can be extended to an automorphism of prime order not coprime to |G|. Another possible direction of further studies would be considering automorphisms of composite order.
In § 3 we perform reduction of the proofs of both Theorems 1.3 and 1.4 to the case of soluble groups, using the classification of finite simple groups. Then in § 4 the proof of Theorem 1.3 about left Engel sinks is completed: first a 'weak' upper bound for the index of F 2 (G) in terms of |ϕ| and m is obtained, and then a 'strong' upper bound, in which the dependence on |ϕ| is eliminated. The proof of Theorem 1.4 is completed in § 5. Requisite preliminary material is collected in § 2.

Preliminaries
For a group A acting by automorphisms on a group B we use the usual notation for commutators [b, a] = b −1 b a and commutator subgroups [B, A] = [b, a] | b ∈ B, a ∈ A , as well as for centralizers C B (A) = {b ∈ B | b a = b for all a ∈ A}. We usually denote by the same letter α the automorphism induced by an automorphism α on the quotient by a normal α-invariant subgroup.
We say for short that an automorphism ϕ of a finite group G is a coprime automorphism if the orders of ϕ and G are coprime: (|G|, |ϕ|) = 1. First we recall some properties of coprime automorphisms of finite groups. Several well-known facts about coprime automorphisms will be often used without special references, but we recall them in the following lemma.  Hall π-subgroup of G. (c) If N is a normal ϕ-invariant subgroup of G, then the fixed points of ϕ in the quotient G/N are covered by the fixed points of ϕ in G, that is, In the next useful lemma the condition that the group is nilpotent cannot be dropped. . Let ϕ be a coprime automorphism of a finite nilpotent group G. Then any element g ∈ G can be uniquely written in the form g = cu, where c ∈ C G (ϕ) and u = [v, ϕ] for some v ∈ G.
As a special case, we have the following well-known fact.
Lemma 2.3. Let ϕ be a coprime automorphism of an abelian finite group G.
The following lemma appeared in [13] in a slightly less general form, so we reproduce the proof for the benefit of the reader.
However, if there are too many steps, then for the element w = u 1 u 2 · · · u k we shall u k ] will have order greater than m, a contradiction.
We now consider the general case. Since every element u ∈ U acts faithfully on [V, u], the exponent of U is m-bounded. If P is a Sylow p-subgroup of U, let M be a maximal normal abelian subgroup of P . By the above, |[V, M]| is m-bounded. Since M acts faithfully on [V, M], we obtain that |M| is m-bounded. Hence |P | is m-bounded, since C P (M) = M and P/M embeds in the automorphism group of M. Since |U| has only m-boundedly many prime divisors, it follows that |U| is m-bounded.
Along with the aforementioned celebrated theorem of Thompson on the nilpotency of finite groups with a fixed-point-free automorphism of prime order, we shall also use the following theorem of Hartley and Meixner about an 'almost fixed-point-free' automorphism.
). If a finite soluble group G admits an automorphism ϕ of prime order p such that |C G (ϕ)| = s, then the index of F (G) is (p, s)-bounded.
We shall use the following lemma on actions of Frobenius groups.
Lemma 2.6 ([12, Lemma 2.4]). Suppose that a finite group G admits a Frobenius group of automorphisms F H with kernel F and complement H such that C G (F ) = 1. Then We denote by γ ∞ (G) = i γ i (G) the intersection of the lower central series of a group G; when G is finite, then G/γ ∞ (G) is the largest nilpotent quotient of G. We note the following elementary fact.
We now recall some elementary properties of Engel sinks. In cases where we need to consider the left or right Engel sink constructed with respect to a subgroup H containing g, we write E H (g) or R H (g), respectively. The following lemma is a consequence of the properties of coprime actions and a formula in Heineken's paper [10] on a connection between left and right Engel commutators.
. If V is an abelian subgroup of a finite group G, and g ∈ G an element normalizing V such that (|V |, |g|) = 1, then We shall freely use this fact without special references.

Bounding the index of the soluble radical
In this section we perform a reduction of the proof of Theorems 1.3 and 1.4 to the case of soluble groups. For that we use a recent result of Guralnick and Tracey [8]. In fact, we can derive from Theorem 3.1 a general proposition about groups of coprime automorphisms whose fixed points have bounded Engel sinks. We state this proposition here in greater generality than needed in the present paper, as it may find applications in other studies. Proof. Since the hypothesis is inherited by G/S(G), we can assume that S(G) = 1. Then the generalized Fitting subgroup is a direct product of non-abelian finite simple groups S i , which are permuted by H. Since the centralizer of F * (G) is trivial and G embeds into the automorphism group of F * (G), it is sufficient to obtain a bound for |F * (G)| in terms of m. We can simply assume that G = F * (G). As proved by Wang and Chen [24] on the basis of the classification of finite simple groups, a finite group admitting a group of coprime automorphisms with fixed-point subgroup of odd order is soluble. Therefore C G (H) contains involutions. Moreover, we can choose an involution α ∈ C G (H) with non-trivial projections onto each of the factors S i in (3.1). Indeed, let {S i 1 , . . . , S i k } be one of the orbits of H in its permutational action on the set {S 1 , . . . , S n }. Clearly, any non-trivial element of the product S i 1 × · · ·× S i k centralized by H must have non-trivial projections onto each of the factors S i j . There is an involution in the centralizer of H in the product S i 1 × · · · × S i k . We now take α ∈ C G (H) to be the product of these involutions over all orbits of H. Then α ∈ C G (H) is an involution with non-trivial projections onto each of the factors S i in (3.1). It follows that G = F * (G) = [G, α].
In our situation, as an obvious consequence we obtain the following. (a) every element g ∈ C G (ϕ) has a left Engel sink E (g) of cardinality at most m, or (b) every element g ∈ C G (ϕ) has a right Engel sink R(g) of cardinality at most m.
Then the soluble radical S(G) has m-bounded index in G.

Completion of the proof for left Engel sinks
Proof of Theorem 1.3. Recall that G is a finite group admitting an automorphism ϕ of prime order p coprime to |G|, and every element of C G (ϕ) has a left Engel sink of cardinality at most m; we need to prove that |G/F 2 (G)| is m-bounded. By Corollary 3.3(a) the index of the soluble radical |G/S(G)| is m-bounded. Therefore we can assume that G is soluble.
First we obtain a 'weak' upper bound for |G/F 2 (G)| in terms of p and m.
Proof. By Gaschütz's theorem the image of the Fitting subgroup F (G) in the quotient G/Φ(G) of G by the Frattini subgroup Φ(G) is the Fitting subgroup of this quotient (see [21, 5.2.15]), and hence the same holds for all terms of the Fitting series. Therefore, since the hypothesis is inherited by G/Φ(G), we can assume from the outset that Φ(G) = 1. Then F (G) is abelian by Gaschütz's theorem.
To lighten the notation, let F i = F i (G). Since F 1 is abelian, every element g ∈ F 1 C F 2 (ϕ) has Engel sink of cardinality at most m. Indeed, let g = uc, where u ∈ F 1 and c ∈ C F 2 (ϕ). For any h ∈ G, a long enough commutator d = [h, k g] belongs to F 1 , since F 2 /F 1 is nilpotent.
Then [d, n g] = [d, n uc] = [d, n c] for any n, since F 1 is abelian. As a result, E (g) is contained in E (c), which has cardinality at most m by hypothesis.
Applying Theorem 1.1, we now obtain that γ ∞ (F 1 C F 2 (ϕ)) has m-bounded order. It follows that the Fitting subgroup F (F 1 C F 2 (ϕ)) has m-bounded index in F 1 C F 2 (ϕ) by Lemma 2.7. But the Fitting subgroup We can apply the same arguments to the quotient of G/F 1 by its Frattini subgroup. We obtain that C F 3 /F 2 (ϕ) also has m-bounded order. As a result, C F 3 /F 1 (ϕ) has m-bounded order. By Theorem 2.5 then the Fitting subgroup F (F 3 /F 1 ) has (p, m)-bounded index in F 3 /F 1 . But, obviously, F (F 3 /F 1 ) = F 2 /F 1 , so that F 3 /F 2 has (p, m)-bounded order. Since G is soluble, F 3 /F 2 contains its centralizer in G/F 2 . Then the group G/F 2 embeds into the automorphism group of F 3 /F 2 and therefore also has (p, m)-bounded order.
We proceed with the proof of Theorem 1.3, where we need to obtain a 'strong' bound, in terms of m only, for |G/F 2 (G)|. If p m, then the bound for |G/F 2 (G)| in terms of p and m obtained in Lemma 4.1 is a required bound in terms of m only. Therefore we can assume that p > m. We observe that when x ∈ C G (ϕ), the Engel sink E (x) is ϕ-invariant. Then the condition p > m |E (x)| implies that for every x ∈ C G (ϕ). (4.1) Let N = C G (ϕ) G be the normal closure of C G (ϕ). Our goal now is to prove that γ ∞ (N) has m-bounded order. The bulk of the proof is in the following key lemma. Proof. We use induction on the order of G. To lighten the notation, we write γ = γ ∞ (N).
Recall that G is soluble. Let V be a minimal normal subgroup of G ϕ , so it is an elementary abelian q-group for some prime q. By induction applied to G/V , we have γ V C G (ϕ). By the minimality of V , either γ ∩ V = 1 or γ V . If γ ∩ V = 1, then [γ, ϕ] [V C G (ϕ), ϕ] = [V, ϕ] ∩ γ = 1, whence γ C G (ϕ) and the proof is complete. Hence we can assume that γ V .
We by (4.1). We record this property in the form As a result, the subgroup O q (γ) generated by all its Hall q ′ -subgroups is contained in C G (ϕ) ∩ γ. If O q (γ) = 1, then by induction γ O q (γ)C G (ϕ) C G (ϕ) and the proof is complete. Therefore we can assume that O q (γ) = 1, that is, γ is a q-group.
Let R be a ϕ-invariant Hall q ′ -subgroup of N. Then [γ, R] = γ and therefore, by since γ is abelian. We now consider γ as a right F q G ϕ -module and extend the ground field to a finite splitting field k of G ϕ ; let M be the resulting kG ϕ -module. The additive group of M is a finite q-group, and therefore both R and ϕ act by coprime automorphisms, to which Lemma 2.1(c) on covering fixed points applies. Note that by (4.3) we have We continue using the same notation for centralizers and commutator subgroups, albeit using the additive structure of M.
We now consider an unrefinable series of kG ϕ -submodules connecting 0 with M. We consider an arbitrary factor U of this series, which is an irreducible kG ϕ -module. Let the bar denote the images of elements and subgroups in the action on U. Note that γ = 1, so thatN is nilpotent. We can sometimes drop the bar when considering the action of G ϕ on U; for example, C U (R) = 0 (4.5) by (4.4) and Lemma 2.1. We also have by (4.2). Let U = W 1 ⊕ · · · ⊕ W n be the decomposition of U into the sum of Wedderburn homogeneous components with respect toR, which is a normal subgroup ofḠ, sinceN is nilpotent. By Clifford's theorem, the group G ϕ transitively permutes the components W i , and the kernel of this permutational action contains RC G (R) N.
We claim that C R (ϕ) acts non-trivially on one of the components W i . Indeed, otherwise C R (ϕ) acts trivially on U. ButR is the normal closure of CR(ϕ) becauseN is nilpotent and is the normal closure of CḠ(ϕ). Thus we obtain that R acts trivially on U, which contradicts (4.5).
For definiteness, let W 1 be a component on which C R (ϕ) acts non-trivially. By (4.6) then [W 1 , C R (ϕ)] (which is contained in W 1 ) is a non-trivial subspace of C U (ϕ). Hence ϕ belongs to the stabilizer of W 1 . SinceN R CḠ(R) is in the kernel of the permutational action on the set of the W i and ϕ acts fixed-point-freely on G/N, all orbits of ϕ on the set of the W i have length p except the one-element orbit {W 1 }. Indeed, let S be the stabilizer of W 1 . If g ∈ S and W 1 gϕ = W 1 g, then, since ϕ ∈ S, the coset Sg is ϕ-invariant. Since |Sg| is coprime to p = |ϕ|, then ϕ would have a fixed point in Sg, contrary to C G (ϕ) N S. We claim that ϕ actually cannot have any orbits of length p on the set of the W i , so that {W 1 } is the only orbit of ϕ, which of course means that U is a homogeneous kRmodule. Suppose the opposite, and let W 1 t, W 1 t ϕ , . . . , W 1 t ϕ p−1 be a regular orbit of ϕ (here, . Then C R (ϕ) must centralize all these W 1 t ϕ i because of (4.6): For every i = 0, 1, . . . , p−1, the subgroup C R (ϕ) t ϕ i acts nontrivially on W 1 t ϕ i and centralizes all the other W 1 t ϕ s for s = i. Let tilde denote the images in the action on the direct sum Then the subgroups C R (ϕ) t , C R (ϕ) t ϕ , . . . , C R (ϕ) t ϕ p−1 commute and generate a direct product in which the factors are permuted by ϕ. The diagonal of this direct product is an image of a subgroup of C R (ϕ) and acts non-trivially on these W t 1 , W t ϕ 1 , . . . , W t ϕ p−1 1 , contrary to (4.7). This contradiction shows that ϕ has no orbits of length p.
Thus, {W 1 } is the only orbit of ϕ, so that U = W 1 is a homogeneous kR-module. SinceR is nilpotent and k is a splitting field, the centre Z(R) acts by scalar multiplications on W 1 . Hence Z(R) commutes withφ. By Lemma 2.1(c) there is an element z ∈ C R (ϕ) whose image is a non-trivial element of Z(R). Since [U, z] C U (ϕ) by (4.6) and U = [U, z], we obtain that [U, ϕ] = 0.
Since the above arguments apply to every irreducible factor of that series in M, we obtain by Lemma 2.1(c) that [M, ϕ] = 0, which means that γ C G (ϕ), as required. Proof. By Lemma 2.2 every element of N/γ ∞ (N) can be written as a product of a commutator [h, ϕ] for h ∈ N/γ ∞ (N) and an element from C N/γ∞(N ) (ϕ) = C N (ϕ)γ ∞ (N)/γ ∞ (N). Since γ ∞ (N) C N (ϕ) by Lemma 4.2, it follows that every element n ∈ N can be written as a product n = [g, ϕ]c for g ∈ N and c ∈ C N (ϕ).
We now claim that E (n) = E (c). This follows from the fact that [g, ϕ] centralizers γ ∞ (N). Indeed, since γ ∞ (N) C N (ϕ) by Lemma 4.2 and γ ∞ (N) is a normal subgroup of G ϕ , the centralizer C G (γ ∞ (N)) is a normal subgroup that contains ϕ and therefore also contains [g, ϕ].
The quotient G/N admits a fixed-point-free automorphism ϕ of prime order and therefore is nilpotent by Thompson's theorem [23]. Therefore for any x ∈ G we have [x, s n] ∈ N for some s, and then [x, s+t n] ∈ γ ∞ (N) for some t. Since [g, ϕ] centralizers γ ∞ (N) as shown above, we further have [x, s+t+u n] = [[x, s+t n], u c] and this element belongs to E (c) for large enough u.
As a result, all elements of N have Engel sinks of size at most m and therefore γ ∞ (N) has m-bounded order by Theorem 1.1.
We now finish the proof of Theorem 1.3. In the remaining case p > m we have γ ∞ (N) of m-bounded order by Lemma 4.3. Then C G (γ ∞ (N)) has m-bounded index and is metanilpo- N)) is nilpotent by Lemma 2.7.

Completion of the proof for right Engel sinks
Proof of Theorem 1.4. Recall that G is a finite group admitting an automorphism ϕ of prime order p coprime to |G|, and every element of C G (ϕ) has a right Engel sink of cardinality at most m; we need to prove that |G/F (G)| is m-bounded. By Corollary 3.3(b) the index of the soluble radical |G/S(G)| is m-bounded. Therefore we can assume that G is soluble.
Since G/F 2 (G) embeds in the automorphism group of F 2 (G)/F (G), we can assume that G = F 2 (G), so that G/F is nilpotent. We choose Thompson's critical subgroup C p in each Sylow p-subgroup P of G/F , which is a characteristic subgroup of P such that C/Z(C) is an elementary abelian p-group and C P (C p ) = Z(C) (see [7,Theorem 5.3.11]). Then the product C = p C p is a characteristic subgroup of G/F such that C/Z(C) is a direct product of elementary abelian groups and C G/F (C) = Z(C). Since (G/F )/Z(C) embeds into the automorphism group of C, it is sufficient to prove that |C| is m-bounded. Therefore we can replace G by the inverse image of C and assume that G/F = C.
Consider the Fitting subgroup F (G ϕ ) of the semidirect product G ϕ . We have F (G ϕ )∩ G = F (G), so it suffices to show that the index of F (G ϕ ) in G ϕ is m-bounded. By Gaschütz's theorem [11,Satz III.4.2] the image of the Fitting subgroup in the quotient of F (G ϕ ) by its Frattini subgroup is the Fitting subgroup of this quotient. Since the hypothesis is inherited by this quotient, we can assume that the Frattini subgroup of F (G ϕ ) is trivial. Then F (G ϕ ) is a direct product of minimal normal subgroups of G ϕ by Gaschütz's theorem [11,Satz III.4.5]. Therefore F (G) is a direct product of minimal normal ϕ-invariant subgroups, which are elementary abelian q-groups for various q. To lighten the notation, in what follows we write F = F (G).
Lemma 5.1. The centralizer C G/F (ϕ) of ϕ in G/F has m-bounded order.
Proof. Let q be any prime dividing |C G/F (ϕ)|, and let Q be a Sylow q-subgroup of C G/F (ϕ). Consider any non-trivial element g ∈ Q, and letĝ be a q-element of G that is a pre-image of g chosen in C G (ϕ) in accordance with Lemma 2.2. Thenĝ induces by conjugation a non-trivial coprime automorphism of the Hall q ′ subgroup F q ′ of F . Indeed, ifĝ centralized F q ′ , thenĝ would centralize all factors of a principal series of G and would belong to F by [21, 5.2.9]. We have [F q ′ ,ĝ] ⊆ R(ĝ) by Lemma 2.8. Hence [F q ′ ,ĝ] = [F q ′ , g] has m-bounded order for any g ∈ Q. Then |Q| is m-bounded by Lemma 2.4. In particular, the prime q is bounded above in terms of m. As a result, |C G/F (ϕ)| is m-bounded.
Due to Lemma 5.1 we can use induction on |C G/F (ϕ)| in the proof of Theorem 1.4 to show that we can assume that C G/F (ϕ) = 1. Namely, if C G/F (ϕ) = 1, then for some prime q and a Sylow q-subgroup Q of G/F , there is a non-trivial element c ∈ C Q (ϕ). As before, [F q ′ , c] = 1. If c ∈ Z(Q), then, since Q/Z(Q) is elementary abelian, by Maschke's theorem there is a normal ϕ-invariant subgroup Q 1 of Q of index q not containing c. Since here q is m-bounded, the inverse image G 1 of the product of Q 1 and the Hall q ′ -subgroup of G/F is a ϕ-invariant subgroup of m-bounded index with F (G 1 ) = F such that |C G 1 /F (ϕ)| < |C G/F (ϕ)|. Then the induction hypothesis applied to G 1 completes the proof. If, however, c ∈ Z(Q), then letĉ be a q-element of G that is a pre-image of c chosen in C G (ϕ) in accordance with Lemma 2.2. Then [F q ′ , c] is a normal ϕ-invariant subgroup of G, since c ∈ Z(G/F ) and F is nilpotent. Furthermore, [F q ′ , c] ⊆ R(ĉ) by Lemma 2.8, and therefore [F q ′ , c] has m-bounded order. We obtain that C G ([F q ′ , c]) is a ϕ-invariant subgroup of m-bounded index, which does not contain c, since [[F q ′ , c], c] = [F q ′ , c] = 1. Then the induction hypothesis applied to C G ([F q ′ , c]) completes the proof.
Thus, we can assume that C G/F (ϕ) = 1 for the rest of the proof. This means that (G/F ) ϕ is a Frobenius group with kernel G/F and complement ϕ acting on F . It follows by Lemma 2.6 that every ϕ-invariant normal section S of G such that C S (G/F ) = 1 is the normal closure of C S (ϕ) under the action of G: Recall that F is a direct product of minimal normal ϕ-invariant subgroups (which are elementary abelian q-groups for various, not necessarily different, primes q): Let V be one of these factors V i , and letḠ = G/C G (V ) be the image of G in the action by conjugation on V . We record some properties of the action of G on V in the following lemma.
Lemma 5.2. If V is an elementary abelian q-group, thenḠ is a non-trivial q ′ -group and C V (G) = 1.
Proof. Indeed, suppose that Q is a nontrivial Sylow q-subgroup ofḠ. SinceḠ is nilpotent, then Z(V Q) ∩ V is a proper G-invariant subgroup of V , which is also ϕ-invariant. This contradicts the minimality of V . Since Z(G) = 1 by our assumption, we haveḠ = 1. Since C V (G) is normal in G ϕ , we must have C V (G) = 1 by the minimality of V .
We now prove a key lemma in the proof of Theorem 1.4. Proof. Let V be an elementary abelian q-group. SinceḠ = 1, we have V = C V (ϕ) g | g ∈ G by (5.1). Let 1 = c ∈ C V (ϕ). For any x ∈Ḡ and any positive integer n we have [c, q n x] = [c, x q n ]. Indeed, regarding V as an F qḠ -module, we see that in characteristic q. For all large enough n the element [c, q n x] belongs to the right Engel sink R(c). Choosing n to be the same large enough integer for all elements x ∈Ḡ, we obtain that [c, x q n ] takes at most m values when x runs overḠ, since |R(c)| m. SinceḠ is a finite q ′ -group by Lemma 5.2, the elements x q n run over the entireḠ as x does. Thus, [c, x] takes at most m values when x runs overḠ, which means that the index of the centralizer CḠ(c) inḠ is at most m. The intersection Z = g∈Ḡ CḠ(c) g is a G-invariant subgroup of m-bounded index, and this intersection is also ϕ-invariant, since CḠ(c) is. Then [c g , Z] = 0 for all g ∈ G. But V = c g | g ∈Ḡ , because V is a minimal normal subgroup of G ϕ , while the right-hand side is both G-and ϕ-invariant. Hence, [V, Z] = 0 or, in other words, Z = 1, so thatḠ has m-bounded order.
We now finish the proof of Theorem 1.4. Recall that any element of G/F acts nontrivially on at least one of the factors V i in (5.2). Let 1 = c 1 ∈ C V 1 (ϕ), which exists by (5.1). Then the right Engel sink of c 1 is non-trivial: R(c 1 ) = {1}. Indeed, otherwise c 1 belongs to the hypercentre ζ ∞ (G) by Baer's theorem [21, 12.3.7], and then V 1 ζ ∞ (G) by the minimality of V 1 . This, however, contradicts Lemma 5.2. Therefore we can choose a 1 ∈ G such that [c 1 , n a 1 ] = 1 for any n ∈ N.
If C G (V 1 ) = F , then |G/F | is m-bounded by Lemma 5.3 and the proof is complete. Otherwise C G (V 1 ) acts non-trivially on at least one of the remaining factors in (5.2). Without loss of generality, we can assume that [V 2 , C G (V 1 )] = 1. Let 1 = c 2 ∈ C V 2 (ϕ), which exists by (5.1). Then the right Engel sink of c 2 in the product V 2 C G (V 1 ) is non-trivial: R V 2 C G (V 1 ) (c 2 ) = {1}. Otherwise c 2 belongs to the hypercentre ζ ∞ (V 2 C G (V 1 )) of the semidirect product V 2 C G (V 1 ) by Baer's theorem [21, 12.3.7], and since V 2 C G (V 1 ) is a normal subgroup of G ϕ , then V 2 ζ ∞ (V 2 C G (V 1 )) by the minimality of V 2 , which contradicts Lemma 5.2. Therefore we can choose a 2 ∈ C G (V 1 ) such that [c 2 , n a 2 ] = 1 for any n ∈ N.
Note that at the same time [c 1 , a 2 ] = 1.
If C G (V 1 V 2 ) = C G (V 1 ) ∩ C G (V 2 ) = F , then |G/F | is m-bounded by Lemma 5.3 and the proof is complete. Otherwise C G (V 1 V 2 ) acts non-trivially on at least one of the remaining factors in (5.2), say on V 3 . Let 1 = c 3 ∈ C V 3 (ϕ), which exists by (5.1). The right Engel sink of c 3 in the product V 3 C G (V 1 V 2 ) is non-trivial: R V 3 C G (V 1 V 2 ) (c 3 ) = {1}, as otherwise c 3 ∈ ζ ∞ (V 3 C G (V 1 V 2 )), whence V 3 ζ ∞ (V 3 C G (V 1 V 2 )) by the minimality of V 3 , contrary to Lemma 5.2. Therefore we can choose a 3 ∈ C G (V 1 V 2 ) such that [c 3 , n a 3 ] = 1 for any n ∈ N.
We can continue this construction in the obvious fashion as long as C G (V 1 V 2 · · · V k ) = F . We claim that this process will stop at C G (V 1 V 2 · · · V k ) = F for some k m. Indeed, otherwise we would have constructed elements c i ∈ V i and a 1 ∈ G and a i ∈ C G (V 1 · · · V i ) for i = 2, . . . , m + 1 such that [c i , n a i ] = 1 for any n ∈ N, while [c i , a j ] = 1 for any j > i.
Then the product c = c 1 c 2 · · · c m+1 would have right Engel sink R(c) of cardinality at least m + 1, since for any given n i ∈ N, i = 1, 2, . . . , m + 1, the m + 1 elements [c, n i a i ] are different. Indeed, as an element of the product V = V 1 × · · · × V k , the commutator [c, n i a i ] has trivial components in the factors V 1 , . . . V i−1 and a non-trivial component in V i . The inequality |R(c)| m + 1 contradicts the hypothesis of the theorem. This contradiction shows that C G (V 1 V 2 · · · V k ) = F for some k m. Then |G/F | = |G/C G (V 1 · · · V k )| = G/ C G (V 1 ) ∩ · · · ∩ C G (V k ) and we obtain that |G/F | is m-bounded by Lemma 5.3.