Profinite groups with an automorphism of prime order whose fixed points have finite Engel sinks
aa r X i v : . [ m a t h . G R ] J a n PROFINITE GROUPS WITH AN AUTOMORPHISM OF PRIME ORDERWHOSE FIXED POINTS HAVE FINITE ENGEL SINKS
E. I. KHUKHRO AND P. SHUMYATSKY
Abstract.
A right Engel sink of an element g of a group G is a set R p g q such that forevery x P G all sufficiently long commutators r ... rr g, x s , x s , . . . , x s belong to R p g q . (Thus, g is a right Engel element precisely when we can choose R p g q “ t u .) We prove that if aprofinite group G admits a coprime automorphism ϕ of prime order such that every fixedpoint of 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 E p g q such that for every x P G allsufficiently long commutators r ... rr x, g s , g s , . . . , g s belong to E p g q . (Thus, g is a left Engelelement precisely when we can choose E p g q “ t u .) We prove that if a profinite group G admits a coprime automorphism ϕ of prime order such that every fixed point of ϕ has afinite 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. Wesay for short that ϕ is a coprime automorphism of G if its order is coprime to the ordersof elements of G (understood as Steinitz numbers), in other words, if G is an inverse limitof finite groups of order coprime to the order of ϕ . Coprime automorphisms of profinitegroups have many properties similar to the properties of coprime automorphisms of finitegroups. 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 bythe same letter) in G { N are images of the fixed points in G , that is, C G { N p ϕ q “ C G p ϕ q N { N .Therefore, if ϕ is a coprime automorphism of prime order p such that C G p ϕ q “
1, Thomp-son’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 coprimeautomorphism of prime order all of whose fixed points are right Engel elements. Recallthat the n -Engel word r y, n x s is defined recursively by r y, x s “ y and r y, i ` x s “ rr y, i x s , x s .An element g of a group G is said to be right Engel if for any x P G there is an integer n “ n p g, x q such that r g, n x s “
1. If all elements of a group are right Engel (therefore alsoleft 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 groupis locally nilpotent. Recall that a group is said to be locally nilpotent if every finite subsetgenerates 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 profinitegroup G . If every element of C G p ϕ q is a right Engel element of G , then G is locally nilpotent .In this paper we consider profinite groups admitting a coprime automorphism of primeorder all of whose fixed points have finite Engel sinks. Recall that Engel sinks are used tostudy generalizations of Engel conditions and are defined as follows. Mathematics Subject Classification . Primary 20E18, 20E36; Secondary 20F19, 20F45.
Key words and phrases . Profinite groups; Engel condition; locally nilpotent; automorphism. efinition. A left Engel sink of an element g of a group G is a set E p g q such that for every x P G all sufficiently long commutators r x, g, g, . . . , g s belong to E p g q , that is, for every x P G there is a positive integer l p x, g q such that r x, l g s P E p g q for all l ě l p x, g q . (Thus, g is a left Engel element precisely when we can choose E p g q “ t u , and G is an Engelgroup when we can choose E p g q “ t u for all g P G .) Definition. A right Engel sink of an element g of a group G is a set R p g q such that forevery x P G all sufficiently long commutators r g, x, x, . . . , x s belong to R p g q , that is, forevery x P G there is a positive integer r p x, g q such that r x, r g s P R p g q for all r ě r p x, g q . (Thus, g is a right Engel element precisely when we can choose R p g q “ t u , and G is anEngel group when we can choose R p g q “ t u for all g P G .)Our main result concerning right Engel sinks is as follows. Theorem 1.2.
Let G be a profinite group admitting a coprime automorphism ϕ of primeorder such that all fixed points of ϕ have finite right Engel sinks . Then G has an open locallynilpotent subgroup .Note that if all elements of a profinite or compact group have finite or even countable leftor 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 thehypotheses 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 Neu-mann’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 earlierin [12]. In that paper [12] we also proved that if a finite group G has a coprime automor-phism ϕ of prime order such that all fixed points of ϕ have left Engel sinks of cardinality atmost m , then G has a metanilpotent subgroup of index bounded in terms of m (examplesshow that here “metanilpotent” cannot be replaced by “nilpotent”). We prove the followingprofinite analogue of this result. Theorem 1.3.
Let G be a profinite group admitting a coprime automorphism ϕ of primeorder p . If all fixed points of ϕ have finite left Engel sinks , then G has an open subgroupthat is an extension of a pronilpotent group by a nilpotent group of class h p p q , where h p p q is Higman’s function depending only on p .There are examples showing that in the conclusion of Theorem 1.3 “pronilpotent-by-nilpotent” cannot be replaced even by “pronilpotent”, in contrast to the stronger virtuallocal nilpotency conclusion of Theorem 1.2 about right Engel sinks. Similarly, if all fixedpoints of ϕ are left Engel elements, then the group G is an extension of a pronilpotent groupby a nilpotent group of class h p p q , where h p p q is Higman’s function (Remark 4.2), but G doesnot have to have an open locally nilpotent subgroup, unlike for the right Engel conditionin Theorem 1.1. Thus, the situation with Engel sinks for fixed points of an automorphismis markedly different from the aforementioned results with conditions on Engel sinks of allelements of a profinite or compact group, where the finiteness (or countability) of right orleft 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 cardinalityat most m , then the conclusions can be strengthened, with bounds for the index of a locallynilpotent (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 § § §
4, respectively. n § Preliminaries
In this section we recall some definitions and general properties related to profinite groupsand 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 x S y . By asubgroup we always mean a closed subgroup, unless explicitly stated otherwise. Recall thatcentralizers are closed subgroups, while commutator subgroups r B, A s “ xr b, a s | b P B, a P A y 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 forcommutators r b, a s “ b ´ b a and commutator subgroups r B, A s “ xr b, a s | b P B, a P A y , aswell as for centralizers C B p A q “ t b P B | b a “ b for all a P A u and C A p B q “ t a P A | b a “ b for all b P B u . A section A { B of a group G is a quotient of a subgroup A ď G by a normalsubgroup B of A . The centralizer of a section is C G p A { B q “ t g P G | r A, g s ď B u . Thedefinition and some properties of coprime automorphisms of profinite groups were alreadymentioned at the beginning of the Introduction in § p group is an inverse limit of finite p -groups, a pronilpotent group is aninverse limit of finite nilpotent groups, a prosoluble group is an inverse limit of finite solublegroups. We denote by π p k q the set of prime divisors of k , where k may be a positive integeror a Steinitz number, and by π p G q 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 π p| g |q Ď σ , and a group G is a σ -group if all of its elements are σ -elements. We denoteby σ the complement of σ in the set of all primes. When σ “ t p u , we write p -element, p -element, etc. Profinite groups have Sylow p -subgroups and satisfy analogues of the Sylowtheorems. Prosoluble groups satisfy analogues of the theorems on Hall π -subgroups. Werefer the reader to the corresponding chapters in [13, Ch. 2] and [19, Ch. 2].We denote by γ p G q “ Ş i γ i p G q the intersection of the lower central series of a group G .A profinite group G is pronilpotent if and only if γ p G q “
1, which is also equivalentto G being the Cartesian product of its Sylow subgroups. Every profinite group G has amaximal normal pronilpotent subgroup denoted by F p G q . This subgroup has the followingcharacterization, similar to that of the Fitting subgroup of a finite group. Lemma 2.1.
The maximal normal pronilpotent subgroup F p G q of a profinite group G isequal to the intersection of the centralizers of all chief factors of all finite quotients of G byopen normal subgroups . Proof . The intersection in question is clearly a closed normal subgroup. In any finite quo-tient of G , the image of this intersection is nilpotent by the well-known characterization ofthe Fitting subgroup of a finite group [14, 5.2.9]. Hence this intersection is contained in F p G q . Conversely, any element of F p G q clearly belongs to the Fitting subgroup of any finitequotient of G and therefore centralizes every chief factor of it. (cid:3) We can define a profinite analogue of the Fitting series by setting F p G q “ F p G q , andthen by induction F k ` p G q being the inverse image of F p G { F k p G qq . It is natural to say thata profinite group has pronilpotent length l if F l p G q “ G and l is minimal with this property.We record a useful elementary lemma about the pronilpotent series. Lemma 2.2. (a) If H is a subgroup of a profinite group G such that F p G q ď H ď F p G q , then F p H q “ F p G q . b) If g is a p -element in F p G qz F p G q , then g induces by conjugation a non-trivial auto-morphism of the Hall p -subgroup of F p G q . Proof . (a) Clearly, F p G q ď F p H q . We now prove the reverse inclusion. Any chief factor A { B of a finite quotient of G is a section of G { F p G q , or of F p G q{ F p G q , or of F p G q . An elementof F p H q centralizes A { B in the first case because H ď F p G q , in the second case because F { F p G q is pronilpotent, and in the third case because F p G q ď H . Hence F p H q ď F p G q by Lemma 2.1.(b) By Lemma 2.1 the element g must act nontrivially on some chief factor of a finitequotient of G by an open normal subgroups. Since g P F p G q , such a chief factor must bea section of F p G q , and since g is contained in a Sylow p -subgroup containing the Sylow p -subgroup of F p G q , such a chief factor must be a section of the Hall p -subgroup of F p G q . (cid:3) If P is a pro- p group, the Frattini subgroup of P is Φ p P q “ r P, P s P p . If α is a co-prime automorphism of P , then α acts nontrivially on P { Φ p P q . The Frattini subgroupof 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 p G { Φ p F p G qqq “ F p G q{ Φ p F p G qq . (2.1) Lemma 2.3.
Let G be a profinite group such that γ p G q is finite . Then C G p γ p G qq is anopen pronilpotent subgroup . Proof . The subgroup C G p γ p G qq is closed and has finite index, since G { C G p γ p G qq faithfullyacts by automorphisms on γ p G q ; hence C G p γ p G qq is an open normal subgroup. Any chieffactor A { B of a finite quotient of G by an open normal subgroup is either a section of G { γ p G q , which is pronilpotent, or of γ p G q . Hence any element of C G p γ p G qq centralizes A { B and the result follows by Lemma 2.1. (cid:3) We recall the well-known consequence of the Baire Category Theorem (see [5, Theo-rem 34]).
Theorem 2.4.
If a profinite group is a countable union of closed subsets , then one of thesesubsets has non-empty interior .We now recall some general properties of Engel sinks. Clearly, the intersection of twoleft Engel sinks of a given element g of a group G is again a left Engel sink of g , with thecorresponding function l p x, g q being the maximum of the two functions. Therefore, if g hasa finite left Engel sink, then g has a unique smallest left Engel sink, which has the followingcharacterization. Lemma 2.5 ([8, Lemma 2.1]) . If an element g of a group G has a finite left Engel sink , then g has a smallest left Engel sink E p g q and for every s P E p g q there is an integer k ě such that s “ r s, k g s .The intersection of two right Engel sinks of a given element g of a group G is again aright Engel sink of g , with the corresponding function r p x, g q being the maximum of the twofunctions. Therefore, if g has a finite right Engel sink, then g has a unique smallest rightEngel sink, which is henceforth denoted by R p g q . 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 R p g q and for every z P R p g q there are integers n ě and m ě and an element x P G such that z “ r g, n x s “ r g, n ` m x s .(Here, the elements x and numbers m, n can be different for different z .)Furthermore, for metabelian groups we have the following. emma 2.7 ([9, Lemma 2.5]) . If G is a metabelian group , then a right Engel sink of theinverse g ´ 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 anopen normal subgroup of G , then Ş p ´ i “ 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 profinitetopology. 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 shalluse this property without special references. The same applies to a subgroup in which everyelement has a finite left Engel sink.Throughout the paper, we write, say, “ p a, b, . . . q -bounded” to abbreviate “bounded abovein terms of a, b, . . . only”. 3. Right Engel sinks
In this section we prove Theorem 1.2 concerning right Engel sinks of fixed points of anautomorphism.
Lemma 3.1. If G is a pronilpotent group and an element g P G has a finite right Engelsink , then in fact R p g q “ t u , that is , g is a right Engel element . Proof . Since R p g q is finite, there is an open normal subgroup N such that R p g q X N “ t u .If z P R p g q , then by Lemma 2.6 there are integers n ě m ě x P G such that z “ r g, n x s “ r g, n ` m x s . Therefore the image of z in G { N must be trivial, since G { N isnilpotent. Hence z P N X R p g q “ t u . (cid:3) 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 ordersuch 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]. Theorem 3.3 ([12, Theorem 1.4]) . Let G be a finite group admitting an automorphism ϕ ofprime order coprime to | G | . Let m be a positive integer such that every element g P C G p ϕ q has a right Engel sink R p g q of cardinality at most m . Then G has a nilpotent normalsubgroup of m -bounded index .In the proof of Theorem 1.2 we will combine this result with Corollary 3.2 and a reductionto the case of uniformly bounded sizes of right Engel sinks of fixed points. Proof of Theorem 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 anopen 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 p ϕ q and let N g be an open normal subgroup such that N g X R p g q “ t u . Then g is a right Engel element of the subgroup N g x g y . By Baer’s theorem [14, 12.3.7], in everyfinite quotient of N g x g y the image of g belongs to the hypercentre. Therefore the subgroup r N g , g s is pronilpotent.Let r N g be the normal closure of r N g , g s in G . Since r N g , g s is normal in the subgroup N g , which has finite index, r N g , g s has only finitely many conjugates. Hence r N g is a prod-uct of finitely many normal subgroups of N g , each of which is pronilpotent, and therefore N g is pronilpotent. Therefore all the subgroups r N g are contained in the largest normalpronilpotent subgroup F p G q .The image ¯ g of every element g P C G p ϕ q in s G “ G { F p G q has finite conjugacy class ¯ g G ,since r g, N g s ď F p G q and N g has finite index in G . We now use a strengthened versionof Neumann’s theorem about BF C -groups and a lemma about finite conjugacy classesin profinite groups from the recent paper of Acciarri and Shumyatsky [1]. Namely, by [1,Lemma 4.2] there is an integer n such that | ¯ g G | ď n for every ¯ g P C s G p ϕ q . Let H “ x C s G p ϕ q G y be the abstract normal closure of C s G p ϕ q in s G . Then by [1, Theorem 1.1] the derived subgroup H is finite (of n -bounded order). In particular, H is a closed subgroup of s G . Let r H bethe topological closure of H in s G . Since H { H is abelian, r H { H is also abelian. (We hadto consider the abstract normal closure first, since [1, Theorem 1.1] is stated for abstractgroups; but it is clear that it also works for profinite groups as shown above.)Note that C s G p ϕ q 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 p G q suchthat s N X H “
1. Then N { F p N q is abelian-by-nilpotent. Replacing G with N we canassume from the outset that G { F p G q is soluble and proceed by induction on the derivedlength of it.The main case is when G { F p G q is abelian. Indeed, in the general case, by inductionhypothesis, G F p G q has a ϕ -invariant open pronilpotent subgroup M . Since G M { M isfinite, there is a ϕ -invariant open normal subgroup N such that N X G “ M . Note that F p N q ě M . Then N { F p N q is abelian, so we may assume that G { F p G q is abelian from theoutset. We need to show that G { F p G q is finite.We write F “ F p G q to lighten the notation. Since F p G { Φ p F qq “ F { Φ p F q by (2.1), wecan assume that Φ p F q “
1. In particular, then G is metabelian. Lemma 3.4. C G p ϕ q F { F is finite . Proof . By Lemma 2.7 we have E p g q Ď R p g ´ q in a metabelian group. Hence all elements of C G p ϕ q have finite left Engel sinks. Every subset E k “ t x P C G p ϕ q | | E p x q| ď k u is closed inthe induced topology of C G p ϕ q . Indeed, this is equivalent to the complement of C G p ϕ qz E k being an open subset of C G p ϕ q . For every element g P C G p ϕ qz E k we have | E p g q| ě k ` z , z , . . . , z k ` P E p g q . By Lemma 2.5 we can write for every i “ , . . . , k ` z i “ r z i , g, . . . , g s , where g is repeated k i ě . (3.1)Let N be an open normal subgroup of G such that the images of z , z , . . . , z k ` are distinctelements in G { N . Then equations (3.1) show that for any u P N X C G p ϕ q the Engel sink E p gu q contains an element in each of the k ` z i N . This means that the wholecoset g p N X C G p ϕ qq is contained in C G p ϕ qz E k . Thus every element of C G p ϕ qz E k has aneighbourhood contained in C G p ϕ qz E k , which is therefore an open subset of C G p ϕ q .Since C G p ϕ q “ Ť k E k , by the Baire Category Theorem 2.4 there is m P N , an open (inthe induced topology) normal subgroup C , and a coset c C such that | E p c x q| ď m forany x P C . We now obtain that | E p x q| is m -bounded for any x P C . Indeed, by [9,Lemma 2.5], in a metabelian group, if E p g q is finite, then E p g q is a normal subgroup. Inthe quotient Ď M “ M { ` E p c q E p c x q ˘ , both Ď M x ¯ c y and Ď M x ¯ c ¯ x y are normal locally nilpotentsubgroups. Hence their product, which contains ¯ x , is also a locally nilpotent subgroup bythe Hirsch–Plotkin theorem [14, 12.1.2]. As a result, E p x q ď E p c q E p c x q and therefore | E p x q| ď | E p c q| ¨ | E p c x q| ď m for any x P C . (3.2) o prove that C G p ϕ q F { F is finite, it remains to show that C F { F is finite. For that weuse the following lemma. Lemma 3.5 ([12, Lemma 2.4]) . Suppose that V is an abelian finite group , and U a groupof coprime automorphisms of V . If |r V, u s| ď n for every u P U , then |r V, U s| is n -bounded , and therefore | U | is also n -bounded .Let q be any prime in π p C F { F q , and let Q be a Sylow q -subgroup of C F . Let f p n q be the function furnished by Lemma 3.5 as a bound in terms of n for | U | . We claim that | QF { F | ď f p m q ; this will imply that C F { F is finite. Note that every element of Q z F acts non-trivially on the Hall q -subgroup F q of F . For every u P Q , the left Engel sink E p u q is a normal subgroup of order ď m by (3.2). Since F q x u y{ E p u q is pronilpotent and u induces a coprime automorphism on F q , we have F q “ E p u q C F q p u q . Hence r F q , u s “r E p u q C F q p u q , u s “ r E p u q , u s “ E p u q , where the last equality holds by the minimality of E p u q . Therefore in every finite quotient s G of G by a ϕ -invariant open normal subgroupwe have | s Q { C s Q p s F q q| ď f p m q by Lemma 3.5. Hence | QF { F | “ | Q { C Q p F q q| ď f p m q , asclaimed.Since C has finite index in C G p ϕ q and C F { F is finite, we conclude that C G p ϕ q F { F isfinite. (cid:3) Let N be a ϕ -invariant open normal subgroup of G containing F such that N X C G p ϕ q ď F .Then Since F p N q “ F , we have C N p ϕ q ď F p N q . Replacing G with N we can assume fromthe outset that C G p ϕ q ď F .Every subset R k “ t x P C G p ϕ q | | R p x q| ď k u is closed in the induced topology of C G p ϕ q . Indeed, this is equivalent to the complement of R k being an open subset of C G p ϕ q .For every element g P C G p ϕ qz R k we have | R p g q| ě k ` z , z , . . . , z k ` P R p g q . Using Lemma 2.6 we can write for every i “ , . . . , k ` z i “ r g, n i x i s “ r g, n i ` m i x s for some x i P G, n i ě , m i ě . (3.3)Let N be an open normal subgroup of G such that the images of z , z , . . . , z k ` are distinctelements in G { N . Then equations (3.3) show that for any u P N X C G p ϕ q the right Engel sink R p gu q contains an element in each of the k ` z i N . Thus, the coset g p N X C G p ϕ qq iscontained in C G p ϕ qz R k . This means that every element of C G p ϕ qz R k has a neighbourhoodcontained in C G p ϕ qz R k , which is therefore an open subset of C G p ϕ q .Since C G p ϕ q “ Ť k R k , by the Baire Category Theorem 2.4 there is m P N , an open (inthe induced topology) normal subgroup C of C G p ϕ q , and a coset c C such that | R p c x q| ď m for any x P C . (3.4)By the standard commutator formula r xy, z s “ r x, z s y r y, z s , using the fact that F isabelian we have r ab, x, . . . , x looomooon n s “ r a, x, . . . , x looomooon n s ¨ r b, x, . . . , x looomooon n s for any a, b P F , any x P G , and any n P N . Therefore we obtain that R p ab q Ď R p a q R p b q for any a, b P C G p ϕ q ď F . The same commutator formula shows that R p a ´ q “ t z ´ | z P R p a qu , so that | R p a ´ q| “ | R p a q| . From (3.4) we now obtain | R p x q| ď | R p c ´ q| ¨ | R p c x q| ď m for any x P C . (3.5)Let N be a ϕ -invariant open normal subgroup of G such that N X C G p ϕ q “ C . Then | R p x q| ď m for any x P C N p ϕ q . By Theorem 3.3, every finite quotient of N by a ϕ -invariantopen normal subgroup has a nilpotent subgroup of index at most f p m q for some function f p n q depending on n alone. Therefore N has a ϕ -invariant open pronilpotent subgroup M of ndex at most f p m q , which is locally nilpotent by Corollary 3.2. Clearly, M is a sought-foropen locally nilpotent subgroup of G . (cid:3) Remark 3.6.
If, under the hypotheses of Theorem 1.2 there is a positive integer n such thatall 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 fromTheorem 3.3 applied to finite quotients of G by a ϕ -invariant open normal subgroup: everysuch quotient has a nilpotent subgroup of index at most f p n q for a function f p n q dependingonly on n . Therefore G has a ϕ -invariant open pronilpotent subgroup M of index at most f p n q , which is locally nilpotent by Corollary 3.2.4. Left Engel sinks
In this section we prove Theorem 1.3 concerning left Engel sinks of fixed points. We beginwith the following lemma.
Lemma 4.1.
Suppose that G “ F p G q is a profinite group of pronilpotent length admittinga coprime automorphism ϕ of prime order such that all elements of C G p ϕ q have finite leftEngel sinks .(a) Then C G { F p G q p ϕ q is finite .(b) If there is m P N such that | E p c q| ď m for all c P C G p ϕ q , then | C G { F p G q p ϕ q| is m -bounded .(c) If all elements of C G p ϕ q are left Engel elements , then C G p ϕ q ď F p G q . Proof . We write F “ F p G q to lighten the notation. Let Φ p F q be the Frattini subgroup of F .Since F p G { Φ p F qq “ F { Φ p F q by (2.1), we can assume that F is abelian in all parts of thelemma.(a) Since the group F C G p ϕ q{ F is pronilpotent and all its elements have finite left Engelsinks, this group is locally nilpotent by [8, Lemma 4.2]. We further claim that all elementsof F C G p ϕ q have finite left Engel sinks in F C G p ϕ q . Indeed, let g “ uc and h “ vd , where u, v P F and c, d P C G p ϕ q . For some k the commutator r h, k g s belongs to F , since x u, v y F { F is nilpotent. Then r h, k ` n g s “ rr h, k g s , n c s for any n , since F is abelian. As a result, E p g q is contained in E p c q , which is finite by hypothesis.Applying [8, Theorem 1.2] we obtain that γ p F C G p ϕ qq is finite. By Lemma 2.3, C G p γ p F C G p ϕ qqq is a closed normal pronilpotent subgroup, which has finite index in F C G p ϕ q . It follows that F p F C G p ϕ qq has finite index in F C G p ϕ q , and the result follows, since F p F C G p ϕ qq “ F byLemma 2.2.(b) If there is m P N such that | E p c q| ď m for all c P C G p ϕ q , then the above argumentshows that | E p g q| ď m for all g P F C G p ϕ q . By [8, Theorem 3.1] then | γ p s F C s G p ϕ qq| is m -bounded for every finite quotient s G of G by a ϕ -invariant open normal subgroup. Hence | γ p F C G p ϕ qq| is also m -bounded, and so is the index of F in F C G p ϕ q by the above argument.(c) Since all elements of C G p ϕ q are left Engel elements, the above argument showsthat C G p ϕ q F is an Engel group, and therefore pronilpotent. Since F “ F p F C G p ϕ qq byLemma 2.2, we obtain F “ F C G p ϕ q . (cid:3) Proof of Theorem G is a profinite group admitting a coprime automorphism ϕ of prime order such that all elements of C G p ϕ q have finite left Engel sinks. We need toshow that G has an open pronilpotent-by-nilpotent subgroup. First we perform reductionto the case of pronilpotent length 3, that is, G “ F p G q . Since all elements of C G p ϕ q havefinite left Engel sinks, by [8, Theorem 1.2] there is a finite normal subgroup C of C G p ϕ q such that C G p ϕ q{ C is locally nilpotent. Let N be a ϕ -invariant open normal subgroup of G such that N X C “
1. Then C N p ϕ q is locally nilpotent. Then the centralizer C s N p ϕ q is ilpotent for every finite quotient s N of N by a ϕ -invariant open normal subgroup. Wangand Chen [18] used the classification of finite simple groups to prove that a finite groupadmitting 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 resultof 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 lengthat most 3. Replacing G with N , we can assume from the outset that G “ F p G q .By Lemma 4.1, both C G { F p G q p ϕ q and C F p G q{ F p G q p ϕ q are finite. Hence, C G { F p G q p ϕ q is finite.Let H be a ϕ -invariant open normal subgroup containing F p G q such that H X C G p ϕ q ď F p G q .Then C H p ϕ q ď F p G q “ F p H q . By the theorems of Thompson [15] and Higman [3] thequotient H { F p H q is nilpotent of class at most h p p q , where p “ | ϕ | .Thus, G has an open subgroup that is an extension of a pronilpotent group by a nilpotentgroup of class at most h p p q . (cid:3) Remark 4.2.
If, under the hypotheses of Theorem 1.3 all fixed points of ϕ are left Engel ele-ments, then the group G is an extension of a pronilpotent group by a nilpotent group of class h p p q , where h p p q is Higman’s function. Indeed, then C G p ϕ q ď F p G q by Lemma 4.1(c). Then G { F p G q is nilpotent of class at most h p p q , where p “ | ϕ | , by the theorems of Thompson [15]and Higman [3]. Remark 4.3.
If, under the hypotheses of Theorem 1.3 there is a positive integer m suchthat all fixed points of ϕ have finite left Engel sinks of cardinality at most m , then the group G has(a) a meta-pronilpotent subgroup of finite m -bounded index, and(b) a subgroup of finite p p, m q -bounded index that is an extension of a pronilpotent groupby a nilpotent group of class g p p q , where p “ | ϕ | and g p p q is a function dependingonly on p .Indeed, then by [12, Theorem 1.3] for every finite quotient s G of G by a ϕ -invariant opennormal subgroup, the index of F p s G q in s G is m -bounded. Hence the index of F p G q in G isalso m -bounded. Furthermore, by Lemma 4.1(b) the order | C ¯ G { F p ¯ G q p ϕ q| is m -bounded. ByKhukhro’s theorem [6, Theorem 2], then ¯ G { F p ¯ G q has a subgroup of p p, m q -bounded indexthat is nilpotent of p -bounded class g p p q . This implies that s G has an open subgroup of p p, m q -bounded index that is nilpotent of class g p p q .5. Examples
Here we present examples showing that in some respects Theorems 1.2 and 1.3 cannot beimproved.
Example 5.1.
Let V be an elementary abelian group of order 7 , and D “ x a y ¸ x b y agroup of automorphisms of V such that a “ b “ C V p a q “
1, and | C V p b q| “
7. Let F “ ś i “ V i be the Cartesian product of isomorphic copies V i of V as F D -modules. Then D can be regarded as a group of automorphisms of F . Let G “ F x a y and let ϕ be theautomorphism of G of order 2 induced by b . Then C G p ϕ q “ ś i “ C V i p ϕ q . Using the factthat all the V i are isomorphic F x a y -modules, one can show that for any c P C G p ϕ q the rightEngel sink R p c q is finite and, moreover, the sizes of these sinks are uniformly bounded. Atthe same time, γ p G q “ F is infinite. This example shows that under the hypotheses ofTheorem 1.2 one cannot obtain a finite subgroup with a pronilpotent quotient. Example 5.2.
For the same V and D “ x a y ¸ x b y as in Example 5.1, let F “ ś ni “ V i bea finite direct product of n copies V i of V as F D -modules. Let G “ F x a y and let ϕ be he automorphism of G of order 2 induced by b . Then C G p ϕ q “ ś ni “ C V i p ϕ q . There is aconstant m independent of n such that | R p c q| ď m for any c P C G p ϕ q . In these examples, γ p G q “ F , so | γ p G q| cannot be bounded in terms of m (and | ϕ | ). This shows that theconclusion of Theorem 3.3 ([12, Theorem 1.4]) also cannot be improved in this respect. Example 5.3.
For the same V and D “ x a y ¸ x b y as in Example 5.1, let H “ ś i “ V i ¸px a i y ¸ x b i yq be the Cartesian product of isomorphic copies V i ¸ px a i y ¸ x b i yq of the semidirectproduct V ¸px a y¸x b yq with V i , a i , b i naturally corresponding to V, a, b . Let G “ ś i “ V i ¸x a i y and let ϕ be the automorphism of G of order 2 induced by the ‘diagonal’ ś i “ b i . Then F p G q “ ś i “ V i and C G p ϕ q “ ś i “ C V i p ϕ q . Since F is abelian, all left Engel sinks of fixedpoints of ϕ are trivial. At the same time, G { F p G q is infinite. This example shows that underthe hypotheses of Theorem 1.3 one cannot obtain an open pronilpotent subgroup (and themore so, a finite normal subgroup with a pronilpotent quotient). Example 5.4.
Similarly to Example 5.2, using finite direct products H “ ś ni “ V i ¸ px a i y ¸x b i yq instead of the Cartesian product, we obtain examples of finite groups G with a coprimeautomorphism ϕ of order 2 such that all elements of C G p ϕ q have trivial left Engel sinks.These examples show that the conclusion of [12, Theorem 1.3] giving a bound for the indexof F p G q cannot be improved to a bound for the index of F p G q . Example 5.5.
Let p be an odd prime and let A “ x a yˆx a y » Z p ˆ Z p be a direct product oftwo copies of p -adic integers regarded as procyclic pro- p groups with (topological) generators a , a . Let B “ x b y » Z p be another procyclic pro- p group with generator b . We define anaction of b by an automorphism on x a y by setting a b “ a p ` . Then we define the action of b by an automorphism on x a y as the inverse of the automorphism a ÞÑ a p ` . The resultingsemidirect product G “ A ¸ B admits an automorphism ϕ of order 2 such that b ϕ “ b ´ and a ϕ “ a . Then C G p ϕ q 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 orderall of whose fixed points have trivial left Engel sinks does not have to have an open locallynilpotent subgroup, in contrast to Theorem 1.1 concerning right Engel sinks. Acknowledgements
The second author was supported by FAPDF and CNPq-Brazil.
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