1 Introduction
Let A be a finite group acting on a finite group G. Many well-known results show that the structure of the centralizer CG(A) (the fixed-point subgroup) of A has influence over the structure of G. The influence is especially strong if (|A|,|G|)=1, that is, the action of A on G is coprime. By following the solution of the restricted Burnside problem, it was discovered that the exponent of CG(A) may have strong impact over the exponent of G. Recall that a group G is said to have exponent n if xn=1 for every x∈G and n is the minimal positive integer with this property. The following theorem was obtained in [14].
Theorem 1.1.
Let q be a prime, n a positive integer and A an elementary abelian group of order q2. Suppose that A acts coprimely on a finite group G and assume that CG(a) has exponent dividing n for each a∈A#. Then the exponent of G is {n,q}-bounded.
Here and throughout the paper A# denotes the set of nontrivial elements of A. Moreover, we will use the expression “{a,b,…}-bounded” to abbreviate “bounded from above in terms of a,b,… only”. The proof of the above result involves a number of deep ideas. In
particular, Zelmanov’s techniques that led to the solution of the restricted Burnside problem [27, 26] are combined with the Lubotzky–Mann theory of powerful p-groups [17, 18],
and a theorem of Bahturin and Zaicev on Lie algebras admitting a solvable group of automorphisms whose fixed-point subalgebra is PI [5].
A profinite (non-quantitative) version of the above theorem was established in [22]. In the context of profinite groups all the usual concepts of group theory are interpreted topologically. In particular, by a subgroup of a profinite group we always mean a closed subgroup and a subgroup is said to be generated by a set S if it is topologically generated by S. By an automorphism of a profinite group we mean a continuous automorphism. We say that a group A acts on a profinite group Gcoprimely if A is finite while G is an inverse limit of finite groups whose orders are relatively prime to the order of A.
The profinite (non-quantitative) version of Theorem 1.1 is as follows.
Theorem 1.2.
Let q be a prime and A an elementary abelian group of order q2. Suppose that A acts coprimely on a profinite group G and assume that CG(a) is torsion for each a∈A#. Then G is locally finite.
In [23] the situation where the centralizers CG(a) consist of Engel elements was dealt with. If x,y are elements of a (possibly infinite) group G, the commutators [x,ny] are defined inductively by the rule
[x,0y]=x,[x,ny]=[[x,n-1y],y] for all n≥1.
An element x is called a (left) Engel element if for any g∈G there exists n, depending on x and g, such that [g,nx]=1. A group G is called Engel if all elements of G are Engel. The element x is called a (left) n-Engel element if for any g∈G we have [g,nx]=1. The group G is n-Engel if all elements of G are n-Engel. The following result was proved in [23].
Theorem 1.3.
Let q be a prime, n a positive integer and A an elementary abelian group of order q2. Suppose that A acts coprimely on a finite group G and assume that for each a∈A# every element of CG(a) is n-Engel in G. Then the group G is k-Engel for some {n,q}-bounded number k.
A profinite (non-quantitative) version of the above theorem was established in the recent work [4].
Theorem 1.4.
Let q be a prime and A an elementary abelian group of order q2. Suppose that A acts coprimely on a profinite group G and assume that all elements in CG(a) are Engel in G for each a∈A#. Then G is locally nilpotent.
A very deep theorem of Wilson and Zelmanov [24, Theorem 5] tells us that a profinite group is locally nilpotent if and only if it is Engel. Thus, there is a clear relation between Theorem 1.3 and Theorem 1.4.
If, in Theorem 1.3, we relax the hypothesis that every element of CG(a) is n-Engel in G and require instead that every element of CG(a) is n-Engel in CG(a), we quickly see that the result is no longer true. An example of a finite non-nilpotent group G admitting an action of a noncyclic group A of
order four such that CG(a) is abelian for each a∈A# can be found for instance in [3]. On the other hand, another result, that was established in [4], is the following.
Theorem 1.5.
Let q be a prime, n a positive integer and A an elementary abelian group of order q3. Suppose that A acts coprimely on a finite group G and assume that for each a∈A# every element of CG(a) is n-Engel in CG(a). Then the group G is k-Engel for some {n,q}-bounded number k.
In [3] a profinite (non-quantitative) version of Theorem 1.5 was obtained. The statement is as follows.
Theorem 1.6.
Let q be a prime and A an elementary abelian group of order q3. Suppose that A acts coprimely on a profinite group G and assume that CG(a) is locally nilpotent for each a∈A#. Then G is locally nilpotent.
The relation between Theorems 1.3 and 1.4 noted above naturally extends to Theorems 1.5 and 1.6.
Let us denote by γi(H) the ith term of the lower central series of a group H and by H(i) the ith term of the derived series of H. It was shown in [9] and further in [1, 2] that if the rank of the acting group A is big enough, then results of similar nature to that of Theorem 1.1 can be obtained while imposing conditions on elements of γi(CG(a)) or CG(a)(i) rather than on elements of CG(a). In the same spirit, one of the goals of the present article is to extend Theorems 1.3 and 1.5, respectively, as follows.
Theorem 1.7.
Let q be a prime, n a positive integer and A an elementary abelian group of order qr with r≥2 acting on a finite q′-group G.
If all elements in γr-1(CG(a)) are n-Engel in G for any a∈A#, then γr-1(G) is k-Engel for some {n,q,r}-bounded number k.
If, for some integer d such that 2d≤r-1, all elements in the dth derived group of CG(a) are n-Engel in G for any a∈A#, then the dth derived group G(d) is k-Engel for some {n,q,r}-bounded number k.
Theorem 1.8.
Let q be a prime, n a positive integer and A an elementary abelian group of order qr with r≥3 acting on a finite q′-group G.
If all elements in γr-2(CG(a)) are n-Engel in CG(a) for any a∈A#, then γr-2(G) is k-Engel for some {n,q,r}-bounded number k.
If, for some integer d such that 2d≤r-2, all elements in the dth derived group of CG(a) are n-Engel in CG(a) for any a∈A#, then the dth derived group G(d) is k-Engel for some {n,q,r}-bounded number k.
Let H,K be subgroups of a profinite group G. We denote by [H,K] the closed subgroup of G generated by all commutators of the form [h,k], with h∈H and k∈K. Thus one can consider inductively the following closed subgroups:
γ1(G)=G,γk(G)=[γk-1(G),G]for k≥1,
G(0)=G,G(k)=[G(k-1),G(k-1)]for k≥1.
Finally, we formulate the (non-quantitative) analogues of Theorems 1.7 and 1.8, respectively.
Theorem 1.9.
Let q be a prime, n a positive integer and A an elementary abelian group of order qr with r≥2 acting coprimely on a profinite group G.
If all elements in γr-1(CG(a)) are Engel in G for any a∈A#, then γr-1(G) is locally nilpotent.
If, for some integer d such that 2d≤r-1, all elements in the dth derived group of CG(a) are Engel in G for any a∈A#, then the dth derived group G(d) is locally nilpotent.
Theorem 1.10.
Let q be a prime, n a positive integer and A an elementary abelian group of order qr with r≥3 acting coprimely on a profinite group G.
If all elements in γr-2(CG(a)) are Engel in CG(a) for any a∈A#, then γr-2(G) is locally nilpotent.
If, for some integer d such that 2d≤r-2, all elements in the dth derived group of CG(a) are Engel in CG(a) for any a∈A#, then the dth derived group G(d) is locally nilpotent.
Thus the purpose of the present article is to provide the proofs for Theorems 1.7, 1.8, 1.9 and 1.10. The paper is organized as follows. In Sections 2 and 3 we present the Lie-theoretic machinery that will be useful within the proofs. Later in Section 4 a technical tool, introduced in [1], is extended to the context of profinite groups. Sections 5 and 6 are devoted to proving Theorems 1.7 and 1.8. Finally, in Section 7 we give the details of the proofs of Theorems 1.9 and 1.10.
Throughout the paper we use, without special references, the well-known properties of coprime actions (see for example [20, Lemma 3.2]):
If α is a coprime automorphism of a profinite group G, then
CG/N(α)=CG(α)N/N
for any α-invariant normal subgroup N.
If A is a noncyclic abelian group acting coprimely on a profinite group G, then G is generated by the subgroups CG(B), where A/B is cyclic.
2 Results on Lie algebras and Lie rings
Let X be a subset of a Lie algebra L. By a commutator in elements of X we mean any element of L that can be obtained as a Lie product of elements of X with some system of brackets. If x1,…,xk,x,y are elements of L, we define inductively
[x1]=x1,[x1,…,xk]=[[x1,…,xk-1],xk]
and
[x,0y]=x,[x,my]=[[x,m-1y],y]
for all positive integers k,m.
As usual, we say that an element a∈L is ad-nilpotent if there exists a positive integer n such that [x,na]=0 for all x∈L. If n is the least integer with the above property, then we say that a is ad-nilpotent of index n.
The next theorem represents the most general form of the Lie-theoretical part of the solution of the restricted Burnside problem. It was announced by Zelmanov in [27]. A detailed proof was published in [28].
Theorem 2.1.
Let L be a Lie algebra over a field and suppose that L satisfies a polynomial identity. If L can be generated by a finite set X such that every commutator in elements of X is ad-nilpotent, then L is nilpotent.
The next theorem, which was proved by Bahturin and Zaicev [5] for soluble groups A and later extended by Linchenko [16] to the general case, provides an important criterion for a Lie algebra to satisfy a polynomial identity.
Theorem 2.2.
Let L be a Lie algebra over a field K. Assume that a finite group A acts on L by automorphisms in such a manner that CL(A) satisfies a polynomial identity. Assume further that the characteristic of K is either 0 or prime to the order of A. Then L satisfies a polynomial identity.
Both Theorems 2.1 and 2.2 admit the following respective quantitative versions (see for example [14] and [21]).
Theorem 2.3.
Let L be a Lie algebra over a field K generated by a1,…,am. Suppose that L satisfies a polynomial identity f≡0 and each commutator in a1,…,am is ad-nilpotent of index at most n. Then the Lie algebra L is nilpotent of {f,K,m,n}-bounded class.
Theorem 2.4.
Let L be as in Theorem 2.2. Assume that CL(A) satisfies a polynomial identity f≡0. Then L satisfies a polynomial identity of {|A|,f,K}-bounded degree.
By combining Theorems 2.3 and 2.4, the following corollary can be obtained.
Corollary 2.5.
Let L be a Lie algebra over a field K and A a finite group of automorphisms of L such that CL(A) satisfies the polynomial identity f≡0. Suppose that the characteristic of K is either 0 or prime to the order of A. Assume that L is generated by an A-invariant set of m elements in which every commutator is ad-nilpotent of index at most n. Then L is nilpotent of {|A|,f,K,m,n}-bounded class.
For our purpose we will need to work with Lie rings, and not only with Lie algebras. As usual, γi(L) denotes the ith term of the lower central series of L. In [23] the following result was established for Lie rings, similar to Corollary 2.5.
Theorem 2.6.
Let L be a Lie ring and A a finite group of automorphisms of L such that
CL(A) satisfies the polynomial identity f≡0. Further, assume that L is generated by an A-invariant set of m elements such that every commutator in the
generators is ad-nilpotent of index at most n. Then there exist positive integers
e and c, depending only on |A|,f,m and n, such that eγc(L)=0.
We also require the following useful lemma whose proof can be found in [14].
Lemma 2.7.
Let L be a Lie ring and H a subring of L generated by m elements h1,…,hm such that all commutators in hi are ad-nilpotent in L of index at most n. If H is nilpotent of class c, then for some {c,m,n}-bounded number u we have
[L,H,…,H⏟u]=0.
Recall that the identity
∑σ∈Sn[y,xσ(1),…,xσ(n)]≡0
is called the linearized n-Engel identity. In general, the statement of Theorem 2.1 cannot be extended to the case where L is just a Lie ring (rather than a Lie algebra over a field). However, such an extension does hold in the particular case where the polynomial identity f≡0 is a linearized Engel identity. More precisely, by combining Theorems 2.6, 2.3 and Lemma 2.7, the following result can be obtained. See [23] for further details.
Theorem 2.8.
Let F be the free Lie ring and f an element of F (Lie polynomial) such that f∉pF for any prime p. Suppose that L is a Lie ring generated by finitely many elements a1,…,am such that all commutators in the generators are ad-nilpotent of index at most n. Assume that L satisfies the identity f≡0. Then L is nilpotent with {f,m,n}-bounded class.
3 On associated Lie rings
Given a group G, there are several well-known ways to associate a Lie ring to it (see [11, 13, 21]). For the reader’s convenience we will briefly describe the construction that we are using in the present paper.
A series of subgroups for G
($*$)G=G1≥G2≥⋯
is called an N-series if it satisfies [Gi,Gj]≤Gi+j for all i,j. Obviously any N-series is central, i.e. Gi/Gi+1≤Z(G/Gi+1) for any i. Given an N-series ($*$), let L*(G) be the direct sum of the abelian groups Li*=Gi/Gi+1, written additively. Commutation in G induces a binary operation [,] in L*(G). For homogeneous elements xGi+1∈Li*,yGj+1∈Lj* the operation is defined by
[xGi+1,yGj+1]=[x,y]Gi+j+1∈Li+j*
and extended to arbitrary elements of L*(G) by linearity. It is easy to check that the operation is well-defined and that L*(G) with the operations + and [,] is a Lie ring.
An N-series ($*$) is called an Np-series if Gip≤Gpi for all i. An important example of an Np-series is the case where the series ($*$) is the p-dimension central series, also known under the name Zassenhaus–Jennings–Lazard series (see [11, p. 250] for details).
Observe that if all quotients Gi/Gi+1 of an N-series ($*$) have prime exponent p, then L*(G) can be viewed as a Lie algebra over the field with p elements.
Any automorphism of G in the natural way induces an automorphism of L*(G). If G is profinite and α is a coprime automorphism of G, then the subring (subalgebra) of fixed points of α in L*(G) is isomorphic to the Lie ring (algebra) associated to the group CG(α) via the series formed by intersections of CG(α) with the series ($*$) (see [21] for further details).
In the case where the series ($*$) is just the lower central series of G we write L(G) for the associated Lie ring. In the case where the series ($*$) is the p-dimension central series of G we write Lp(G) for the subalgebra generated by the first homogeneous component G1/G2 in the associated Lie algebra over the field with p elements.
Let H be a subgroup of G. For a series ($*$) we write
L*(G,H)=⊕j≥1(Gj∩H)Gj+1Gj+1
and, if the series ($*$) is the p-dimension central series, we write
Lp(G,H)=Lp(G)∩L*(G,H).
In particular, if a group A acts coprimely on G, then we have
L*(G,CG(A))=CL*(G)(A) and Lp(G,CG(A))=CLp(G)(A).
We will also require the following lemma that essentially is due to Wilson and Zelmanov (cf. [24, Lemma in Section 3]).
Lemma 3.1.
Let G be a profinite group and g∈G an element such that for any x∈G there exists a positive n with the property that [x,gn]=1. Let L*(G) be the Lie algebra associated with G using an Np-series ($*$) for some prime p. Then the image of g in L*(G) is ad-nilpotent.
We close this section by quoting the following result
whose proof can be found in [14].
Theorem 3.2.
Let P be a d-generated finite p-group and suppose that the Lie algebra Lp(G) is nilpotent of class c. Then P has a powerful characteristic subgroup of {p,c,d}-bounded index.
Recall that powerful p-groups were introduced by Lubotzky and Mann [17, 18]. A finite p-group P is said to be powerful if and only if [P,P]≤Pp for p≠2 (or [P,P]≤P4 for p=2), where Pi denotes the subgroup of P generated by all ith powers. Powerful p-groups have some nice properties. In particular, if P is a powerful p-group, then the subgroups γi(P),P(i) and Pi are also powerful. Moreover, for given positive integers n1,…,ns, it follows, by repeated applications of [17, Propositions 1.6 and 4.1.6], that
[Pn1,…,Pns]≤γs(P)n1⋯ns. Furthermore, if a powerful p-group P is generated by d elements, then any subgroup of P can be generated by at most d elements and P is a product of d cyclic subgroups, that is, P has cyclic subgroups C1,…,Cd with the property that for every element x∈P there exist x1∈C1,…,xd∈Cd such that x=x1⋯xd.
4 On a technical tool: A-special subgroups
The main step in order to deal with the proof of part (2) of Theorems 1.7 and 1.8 is to consider the case where G is a p-group, which can be treated via Lie methods. Then the general case will follows from a reduction to the case of p-groups. We will deal with the case of p-groups by combining Lie methods with the use of the technical concept of A-special subgroups of a group G. This concept was introduced in [1]. In what follows, we are going to provide the reader with the most relevant information on that topic. Let us start by recalling the definition.
Definition 4.1.
Suppose that A is an elementary abelian q-group acting on a finite q′-group G. Let A1,…,As be the maximal subgroups of A and H a subgroup of G.
We say that H is an A-special subgroup of G of degree 0 if and only if H=CG(Ai) for suitable i≤s.
Suppose that k≥1 and the A-special subgroups of G of degree k-1 are defined. Then H is said to be an A-special subgroup of G of degree k if and only if there exist A-special subgroups J1 and J2 of G of degree k-1 such that H=[J1,J2]∩CG(Ai) for suitable i≤s.
Note that the A-special subgroups of G of any degree are A-invariant. If A has order qr, then for a given integer k the number of A-special subgroups of G of degree k is bounded in terms of q,r and k. Moreover, the A-special subgroups have nice properties that are crucial for our purpose. We state here some of those properties whose proofs can be found in [1, Proposition 3.2, Theorem 4.1 and Corollary 4.4].
Proposition 4.2.
Let A be an elementary abelian q-group of order qr with r≥2 acting on a finite q′-group G and let A1,…,As be the maximal subgroups of A. Let k≥0 be an integer.
If k≥1, then every A-special subgroup of G of degree k is contained in some A-special subgroup of G of degree k-1.
If 2k≤r-1 and H is an A-special subgroup of G of degree k, then H is contained in the kth derived group of CG(B) for some subgroup B≤A such that |A/B|≤q2k.
Let Rk be the subgroup generated by all A-special subgroups of G of degree k. Then Rk=G(k).
Let H be an A-special subgroup of G. If N is an A-invariant normal subgroup of G, then the image of H in G/N is an A-special subgroup of G/N.
Theorem 4.3.
Let A be an elementary abelian q-group of order qr with r≥2 acting on a finite q′-group G. Let p be a prime and P an A-invariant Sylow p-subgroup of G(d), for some integer d≥0. Let P1,…,Pt be the subgroups of the form P∩H, where H ranges through A-special subgroups of G of degree d. Then P=P1⋯Pt.
In order to deal with the statements (1) of Theorems 1.7 and 1.8, we will need to use the concept of γ-A-special subgroups of a group G, whose definition was also given in [1]. They are analogues to the A-special subgroups defined above, and their definition is more suitable to treat situations involving the terms of the lower central series of a group. The definition is as follows.
Definition 4.4.
Suppose that A is an elementary abelian q-group acting on a finite q′-group G. Let A1,…,As be the maximal subgroups of A and H a subgroup of G.
We say that H is a γ-A-special subgroup of G of degree 1 if and only if H=CG(Ai) for suitable i≤s.
Suppose that k≥2 and the γ-A-special subgroups of G of degree k-1 are defined. Then H is said to be a γ-A-special subgroup of G of degree k if and only if there exist a γ-A-special subgroup J of G of degree k-1 such that H=[J,CG(Ai)]∩CG(Aj) for suitable i,j≤s.
The next results are similar to Proposition 4.2 and Theorem 4.3, respectively, and their proofs can be found in [1].
Proposition 4.5.
Let A be an elementary abelian q-group of order qr with r≥2 acting on a finite q′-group G and A1,…,As the maximal subgroups of A. Let k≥1 be an integer.
If k≥2, then every
γ
-A-special subgroup of G of degree k is contained in some
γ
-A-special subgroup of G of degree k-1.
If k≤r-1 and H is a
γ
-A-special subgroup of G of degree k, then one has H≤γk(CG(B)) for some subgroup B≤A such that |A/B|≤qk.
Let Rk be the subgroup generated by all
γ
-A-special subgroups of G of degree k. Then Rk=γk(G).
Let H be a
γ
-A-special subgroup of G. If N is an A-invariant normal subgroup of G, then the image of H in G/N is a
γ
-A-special subgroup of G/N.
Theorem 4.6.
Let A be an elementary abelian q-group of order qr with r≥2 acting on a finite q′-group G. Let p be a prime and let P be an A-invariant Sylow p-subgroup of γr-1(G). Let P1,…,Pt be all the subgroups of the form P∩H where H ranges through γ-A-special subgroups of G of degree r-1. Then P=P1⋯Pt.
Since Theorems 1.9 and 1.10 are (non-quantitative) profinite versions of Theorems 1.7 and 1.8, respectively, it is natural to expect that the main step of their proofs is to consider the case of pro-p groups, which
will be treated by using Lie methods. For our purpose we also need to extend the concepts of A-special and γ-A-special subgroups to profinite groups.
Let H and K be subgroups of a profinite group G. Recall that we denote by [H,K] the closed subgroup of G generated by all commutators of the form [h,k], with h∈H and k∈K. In the same spirit of what was done in [1] for the finite case, we can define the concept of A-special subgroups for a profinite group as follows.
Definition 4.7.
Let A be an elementary abelian q-group acting coprimely on a profinite group G. Let A1,…,As be the maximal subgroups of A and H a subgroup of G.
We say that H is an A-special subgroup of G of degree 0 if and only if H=CG(Ai) for suitable i≤s.
Suppose that k≥1 and the A-special subgroups of G of degree k-1 are defined. Then H is said to be an A-special subgroup of G of degree k if and only if there exist A-special subgroups J1 and J2 of G of degree k-1 such that H=[J1,J2]∩CG(Ai) for suitable i≤s.
Note that combining the definition above with a standard inverse limit argument and the results obtained in [1], it is easy to show that A-special subgroups of a profinite group satisfy properties analogous to those listed in Proposition 4.2. Moreover, a profinite version of Theorem 4.3 holds.
In order to deal with part (1) of Theorems 1.9 and 1.10, we need to introduce γ-A-special subgroups of a profinite group. This is done by slightly modifying Definition 4.7 in a similar way to what is stated in Definition 4.4 for the finite case. As a consequence we obtain that analogous profinite versions of Proposition 4.5 and Theorem 4.6 can be established. We omit further details.
5 Proof of Theorem 1.7
Our goal here is to prove Theorem 1.7.
First of all we need to establish the following result about associated Lie rings.
Proposition 5.1.
Let G be a finite group satisfying the hypothesis of part (2) in Theorem 1.7. Suppose that there exists an A-invariant p-subgroup H of G(d), with p a prime divisor of the order of G(d), such that H=H1⋯Ht, where each subgroup Hi is contained in some A-special subgroup of G of degree d and H is generated by a {q,r,t}-bounded number of elements. Then:
Lp(H) is nilpotent of {n,p,q,r,t}-bounded class.
There exist positive integers e and c, depending only n,q,r and t, such that eγc(L(H))=0.
We now deal with the proof of the second statement of Proposition 5.1.
Proof.
Let L=L(H) be the Lie ring associated with the p-subgroup H of G(d). Denote by V1,…,Vt the images of H1,…,Ht in H/γ2(H). It follows that the Lie ring L is generated by V1,…,Vt.
Since H is A-invariant, the group A acts on L in the natural way.
Let A1,…,As be the distinct maximal subgroups of A.
Let W be an additive subgroup of L. We say that W is a special subspace of weight 1 if and only if W=Vj for some j≤t and say that W is a special subspace of weight α≥2 if W=[W1,W2]∩CL(Al), where W1 and W2 are some special subspaces of L of weight α1 and α2 such that α1+α2=α and Al is some maximal subgroup of A for a suitable l≤s.
We claim that every special subspace W of L corresponds to a subgroup of an A-special subgroup of G of degree d. We argue by induction on the weight α of W. If α=1, then W=Vj and so W corresponds to Hj for some j≤t. Assume that α≥2 and write W=[W1,W2]∩CL(Al). By induction we know that W1 and W2 correspond respectively to some J1 and J2 which are subgroups of some A-special subgroups of G degree d.
Note also that [W1,W2] is contained in the image of
[J1,J2]. This implies that the special subspace W corresponds to a subgroup of [J1,J2]∩CG(Al) which, by Proposition 4.2 (1), is contained in some A-special subgroup of G of degree d, as claimed.
Moreover, it follows from Proposition 4.2 (2) that every element of W corresponds to some element of CG(a)(d) for a suitable a∈A#.
Therefore, since all elements of CG(a)(d) are n-Engel in G, we have:
Claim 5.1.
Every element of W is ad-nilpotent of index at most n.
From the previous argument we deduce that L=〈V1,…,Vt〉 is generated by ad-nilpotent elements of index at most n, but we do not know whether every Lie commutator in these generators is again in some special subspace of L and hence it is ad-nilpotent of bounded index.
In order to overcome this problem, we take a qth primitive root of unity ω and put L¯=L⊗ℤ[ω]. We regard L¯ as a Lie ring and remark that there is a natural embedding of the ring L into the ring L¯. In what follows we write V¯ to denote V⊗ℤ[ω], for some subspace V of L.
Let W be a special subspace of L. We claim the following:
Claim 5.2.
There exists an {n,q}-bounded number u such that every element w of W¯ is ad-nilpotent of index at most u.
Indeed, choose w∈W¯ and write
w=l0+ωl1+⋯+ωq-2lq-2,
for suitable elements l0,…,lq-2 of W which in particular correspond to some elements x0,…,xq-2 of CG(a)(d), for a suitable a∈A#.
Denote by
K=〈l0,ωl1,…,ωq-2lq-2〉
the subring of L¯ generated by
l0,ωl1,…,ωq-2lq-2; put H0=〈x0,…,xq-2〉. We will show that K is nilpotent of {n,q}-bounded class. Note that L(H,H0) satisfies the linearized n-Engel identity and, so, the same identity is also satisfied in L(H,H0)¯ which contains K. Observe that
a commutator in the elements l0,ωl1,…,ωq-2lq-2 is of the form ωαv for some v∈W, and so, by Claim 5.1, it is ad-nilpotent of index at most n. Hence, by Theorem 2.8, K is nilpotent of {n,q}-bounded class. Now Lemma 2.7
ensures that there exists an integer u>0, depending only on n and q, such that [L¯,uK]=0. Since w∈K, we conclude that w is ad-nilpotent in L¯ with {n,q}-bounded index, as claimed in Claim 5.2.
The group A acts on L¯ in the natural way.
An element x∈L¯ will be called a common “eigenvector” for A if for any a∈A# there exists a number λ such that xa=ωλx. Since (|A|,|G|)=1 and H can be generated by a {q,r,t}-bounded number of elements, we can choose elements v1,…,vτ in V1¯∪⋯∪Vt¯, that generated the Lie ring L¯, where τ is a {q,r,t}-bounded number, and each vi is a common eigenvector for A (see for example [13, Lemma 4.1.1]).
Let v be any Lie commutator in v1,…,vτ. We wish to show that v belongs to some W¯, where W is a special subspace of L. We argue by induction on the weight of v. If v has weight 1, there is nothing to prove. Assume that v has weight at least 2. Write v=[w1,w2] for some w1∈W1¯ and w2∈W2¯, where W1 and W2 are two special subspace of L of smaller weights. It is clear that v belongs to [W1¯,W2¯]. Note that any commutator in common eigenvectors is again a common eigenvector for A. Therefore v is a common eigenvector and it follows that there exists some maximal subgroup Al of A such that v∈CL¯(Al). Thus v∈[W1¯,W2¯]∩CL¯(Al). Hence v lies in W¯, where W is the special subspace of L of the form [W1,W2]∩CL(Al) and so by Claim 5.2, v is ad-nilpotent of index at most u. This proves the following:
Claim 5.3.
Any commutator in v1,…,vτ is ad-nilpotent of index at most u.
Note that for any a∈A#, the centralizer CL(a)=L(H,CH(a)) satisfies the linearized version of the identity
[y,nδ2d(y1,…,y2d)]≡0,
where δi(y1,…,y2i) is given recursively by
δ0(y1)=y1,
δi(y1,…,y2i)=[δi-1(y1,…,y2i-1),δi-1(y2i-1+1,…,y2i)]
for any i≥1.
The same identity also holds in CL¯(a)=CL(a)¯. Thus, by Theorem 2.6, there exist positive integers e and c depending only on n,q,r and t such that eγc(L¯)=0. Since L embeds into L¯, we also have eγc(L)=0, as desired.
∎
Note that the proofs of items (1) and (2) of the previous proposition are very similar. As for the proof of item (1), we only observe that it can be obtained, with obvious changes, simply by replacing every appeal to Theorem 2.6 in the proof of (2) by an appeal to Corollary 2.5.
We are now ready to embark on the proof of part (2) of Theorem 1.7.
Proof.
By Proposition 4.2 (3) we know that G(d) is generated by A-special subgroups of G of degree d and Proposition 4.2 (2) tells us that any A-special subgroup of G of degree d is contained in CG(B)(d) for some suitable nontrivial subgroup B≤A such that |A/B|≤qr-1. Thus each A-special subgroup of G of degree d is contained in CG(a)(d) for some suitable a∈A#. This implies that G(d) is generated by n-Engel elements. Hence by Baer’s Theorem [10, Theorem III 6.14] we get that G(d) is nilpotent. Then G(d) is a direct product of its Sylow subgroups.
Let π(G(d)) be the set of prime divisors of |G(d)|.
Choose now p∈π(G(d)) and let P be the Sylow p-subgroup of G(d). By Theorem 4.3, P=P1⋯Pt, where each Pi is of the form P∩H for some A-special subgroup H of G of degree d. Combining this with Proposition 4.2 (2), we see that each Pi is contained in CG(a)(d), for some a∈A#. Furthermore, t is a {q,r}-bounded number.
Choose arbitrary elements x,y∈P. Let us write x=x1⋯xt, y=y1⋯yt, where xi and yi belong to Pi. In what follows we will show that 〈x,y〉 is nilpotent of {n,q,r}-bounded class. Let Y be the subgroup generated by the orbits xiA and yiA for i=1,…,t. Note that Y is generated by a {q,r}-bounded number of elements. Since the subgroup 〈x,y〉 is contained in Y, it is enough to show that Y is nilpotent of {n,q,r}-bounded class.
Set Yi=Pi∩Y, i=1,…,t, and note that every Yi is a subgroup of CG(a)(d) for a suitable a∈A#. Since we have Y=〈xiA,yiA:i=1,…,t〉 and every Pi is an A-invariant subgroup, it follows that Y=〈Y1,…,Yt〉. By [1, Lemma 2.1]
we see that Y=Y1⋯Yt. Moreover, note that Y is generated by a {q,r}-bounded number of elements which are
n-Engel.
Now by Proposition 5.1 (2) there exist integers e and c, that depend only on n,q and r, such that eγc(L(Y))=0. If p is not a divisor of e, then we have γc(L(Y))=0 and so Y is nilpotent of class at most c-1. In that case Y is nilpotent of {n,q,r}-bounded class and, in particular, the same holds for 〈x,y〉. Assume now that p is a divisor of e. By Proposition 5.1 (1) we know that Lp(Y) is nilpotent of {n,q,r}-bounded class. Now Theorem 3.2 tells us that Y has a powerful characteristic subgroup K of {n,q,r}-bounded index. It follows from [19, Theorem 6.1.8 (ii), p. 164] that K has an {n,q,r}-bounded rank.
Put R=Ke and assume that R≠1. Note that, if p≠2, then we have
[R,R]≤[K,K]e2≤Kpe2=Rpe,
and if p=2, then we have
[R,R]≤R4e.
Since eγc(L(R))=0, we get that
γc(R)e≤γc+1(R).
Taking into account that R is powerful, we obtain, if p≠2, that
γc(R)e≤γc+1(R)=[R′,c-1R]≤[Rpe,c-1R]≤γc(R)pe
and, if p=2, that
γc(R)e≤γc(R)4e.
Hence we have γc(R)e=1. Since γc(R) is also powerful and generated by an {n,q,r}-bounded number of elements, we infer that γc(R) is of {n,q,r}-bounded order, since it is a product of an {n,q,r}-bounded number of cyclic subgroups. It follows that R has an {n,q,r}-bounded derived length. Recall that R=Ke and K is a powerful p-group. Thus, K has {n,q,r}-bounded derived length and this implies that the derived length of Y is {n,q,r}-bounded, as well.
Now [23, Lemma 4.1] tells us that Y has {n,q,r}-bounded nilpotency class, and the same holds for 〈x,y〉, as desired.
From the argument above we deduce that each Sylow p-subgroup of G(d) is k-Engel, for some {n,q,r}-bounded number k. The result follows.
∎
We conclude this section observing that the proof of part (1) of Theorem 1.7 has a very similar structure to that of part (2). We will omit details and describe only main steps that are somewhat different from those of part (2). More precisely, the first step consists in proving the following analogue of Proposition 5.1.
Proposition 5.2.
Let G be a finite group satisfying the hypothesis of part (1) in Theorem 1.7. Suppose that there exists an A-invariant p-subgroup H of γr-1(G), with p a prime divisor of the order of γr-1(G), such that H=H1⋯Ht, where each subgroup Hi is contained in some γ-A-special subgroup of G of degree r-1 and H is generated by a {q,r,t}-bounded number of elements. Then:
Lp(H) is nilpotent of {n,p,q,r,t}-bounded class.
There exist positive integers e and c, depending only n,q,r and t, such that eγc(L(H))=0.
Next, one can establish part (1) of Theorem 1.7 by replacing every appeal to Theorem 4.3 and Proposition 4.2 in the proof of item (2) by an appeal to Theorem 4.6 and Proposition 4.5, respectively.
6 Proof of Theorem 1.8
In this section we are concerned with the proof of Theorem 1.8. In parallel to what we did in the previous section, we will focus our attention on the proof of statement (2) of Theorem 1.8.
First of all, we will require the following analogue of Proposition 5.1.
Proposition 6.1.
Let G be a finite group satisfying the hypothesis of part (2) in Theorem 1.8. Suppose that there exists an A-invariant p-subgroup H of G(d), with p a prime divisor of the order of G(d), such that H=H1⋯Ht, where each subgroup Hi is contained in some A-special subgroup of G of degree d and H is generated by a {q,r,t}-bounded number of elements. Then:
Lp(H) is nilpotent of {n,p,q,r,t}-bounded class.
There exist positive integers e and c, depending only n,q,r and t, such that eγc(L(H))=0.
In what follows, we outline the proof of item (2) of Proposition 6.1. The proof of part (1) can be obtained with a similar argument.
Proof.
By hypothesis we know that each subgroup Hi for i=1,…,t, is contained in some A-special subgroup of G of degree d. Hence, Proposition 4.2 (2) implies that each Hi is contained in CG(B)(d), for some subgroup B of A such that |A/B|≤q2d. Let A1,…,As be the maximal subgroups of A. For any Aj the intersection B∩Aj is not trivial. Thus, there exists a∈A# such that the centralizer CG(Aj) is contained in CG(a) and Hi is contained in CG(a)(d). Since all elements of CG(a)(d) are n-Engel in CG(a), we deduce:
Claim 6.1.
Each element h∈Hi is n-Engel in CG(Aj), for any j≤s.
We now consider L=L(H) the Lie ring associated to the p-subgroup H of G(d). In the same spirit of what we did in the proof of Proposition 5.1 we define special subspaces of L for any weight and observe that every element of a special subspace W of L corresponds to an element of a subgroup of an A-special subgroup of G of degree d. Since 2d≤r-2, it follows that L=∑j≤sCL(Aj). Now taking into account that every special subspace W of L is contained in some L(H,Hi) and that, by Claim 6.1, we have [CL(Aj),nl]=0 for any l∈L(H,Hi), we deduce that any element of a special subspace W is ad-nilpotent of index at most n in L.
The rest of the proof consists in mimicking the argument used in the proof of Proposition 5.1, with only obvious changes, so we omit the further details.
∎
Now we are ready to deal with the proof of part (2) of Theorem 1.8.
Proof.
By the well-known Zorn’s Theorem [10, Theorem III 6.3] each CG(a)(d) is nilpotent. Furthermore,
[2, Theorem 31] implies that G(d) is nilpotent. It follows that G(d) is a direct product of its Sylow subgroups.
Choose p∈π(G(d)) and let P be a Sylow p-subgroup of G(d).
By Theorem 4.3 we know that P=P1⋯Pt, where t is a {q,r}-bounded number and each subgroup Pi is of the form P∩H, where H is some A-special subgroup of G of degree d.
Moreover, Proposition 4.2 (2) tells us that each Pi≤CG(B)(d), for some subgroup B of A such that |A/B|≤q2d.
Let A1,…,As be the maximal subgroups of A. For any Aj the intersection B∩Aj is not trivial. Thus, there exists a∈A# such that the centralizer CG(Aj) is contained in CG(a) and Pi is contained in CG(a)(d). Thus:
Claim 6.2.
Each element l∈Pi is n-Engel in CG(Aj), for any j≤s.
Choose arbitrary elements x,y∈P. We will show that the subgroup 〈x,y〉 is nilpotent of {n,q,r}-bounded class.
Following an argument similar to that used in the proof of Theorem 1.7 (2), we write x=x1⋯xt and y=y1⋯yt, where each xi and yi belongs to Pi, for i=1,…,t, and want to show that the subgroup Y=〈xiA,yiA:i=1,…,t〉 is nilpotent of {n,q,r}-bound class. Appealing to Proposition 6.1 and Theorem 3.2 we find that Y has a powerful characteristic subgroup N of {n,q,r}-bounded index and, by [19, Theorem 6.1.8 (ii), p. 164], of {n,q,r}-bounded rank as well.
It follows that each CN(a)(d) is an n-Engel subgroup and can be generated by an {n,q,r}-bounded number of elements. Zelmanov noted in [25] that the nilpotency class of a finite n-Engel group is bounded in terms of n and the number of generators of that group. We conclude that each CN(a)(d) is nilpotent of {n,q,r}-bounded class. Now [2, Theorem 31] tells us that N(d) is nilpotent of {n,q,r}-bounded class, and this implies that Y has {n,q,r}-bounded derived length l, say.
By [23, Lemma 4.1] it is enough to see that there exists an {n,q,r}-bounded number u such that each generator of Y is an u-Engel element in Y. Indeed, let M=Y(l-1) be the last nontrivial term of derived series of Y. By induction on l we see that Y/M is of {n,q,r}-bounded nilpotency class, say c1. For any generator x of Y we have
[Y,x,…,x⏟c1]⊆M.
Since M is abelian and A-invariant, we can write
M=∑j≤sCM(Aj).
By Claim 6.2 we have
[M,x,…,x⏟n]=1.
Thus,
[Y,x,…,x⏟c1+n]=1.
This implies that 〈x,y〉 is nilpotent of {n,q,r}-bound class, as desired.
By the argument above we obtain that each Sylow p-subgroup P of G(d) is k-Engel for some {n,q,r}-bounded number k. The result follows.
∎
We finish by noting that the proof of item (1) of Theorem 1.8 can be obtained by replacing every appeal to
[2, Theorem 31], Theorem 4.3, Proposition 4.2 and Proposition 6.1 in the proof of (2) by an appeal to [2, Theorem 41], Theorem 4.6, Proposition 4.5 and the following analogue of Proposition 6.1, respectively.
Proposition 6.2.
Let G be a finite group satisfying the hypothesis of part (1) in Theorem 1.8. Suppose that there exists an A-invariant p-subgroup H of γr-2(G), with p a prime divisor of the order of γr-2(G), such that H=H1⋯Ht, where each subgroup Hi is contained in some γ-A-special subgroup of G of degree r-2 and H is generated by a {q,r,t}-bounded number of elements. Then:
Lp(H) is nilpotent of {n,p,q,r,t}-bounded class.
There exist positive integers e and c, depending only n,q,r and t, such that eγc(L(H))=0.
7 Results on profinite groups
In this section we deal with the proofs of Theorems 1.9 and 1.10 which are profinite non-quantitative analogues of Theorems 1.7 and 1.8, respectively.
Let w=w(x1,x2,…,xk) be a group-word. Let H be a subgroup of a group G and g1,g2,…,gk∈G. We say that the law w≡1 is satisfied on the cosets g1H,g2H,…,gkH if w(g1h1,g2h2,…,gkhk)=1 for all h1,h2,…,hk∈H. Let us start with a lemma.
Lemma 7.1.
Assume that a finite group A acts coprimely on a profinite group G. Then for each prime p the following holds:
If, for some integer k, all elements in γk(CG(A)) are Engel in CG(A), then Lp(G) satisfies a multilinear Lie polynomial identity.
If, for some integer k, all elements in the kth derived group of CG(A) are Engel in CG(A), then Lp(G) satisfies a multilinear Lie polynomial identity.
The proofs of items (1) and (2) of the lemma are similar, so we give a detailed proof of the second statement.
Proof.
Let L=Lp(G).
In view of Theorem 2.2, it is sufficient to show that CL(A) satisfies a polynomial identity. We know that CL(A) is isomorphic to the Lie algebra associated with the central series of CG(A) obtained by intersecting CG(A) with the p-dimension central series of G.
Let
T=CG(A)×⋯×CG(A)⏟2k+1.
For each integer i we define the set
Si={(t,t1,…,t2k)∈T:[t,iδk(t1,…,t2k)]=1}.
Since the sets Si are closed in T and their union coincides with T, by the Baire category theorem [12, p. 200] at least one of these sets has a non-empty interior. Therefore, we can find an open subgroup H in CG(A), elements g,g1,…,g2k in CG(A) and an integer n such that the identity [x,nδk(y1,…,y2k)]≡1 is satisfied on the cosets gH,g1H,…,g2kH. Thus, the Wilson–Zelmanov result [24, Theorem 1] tells us that CL(A) satisfies a polynomial identity.
∎
We will also require the following profinite version of [1, Lemma 2.1]. The proof is straightforward so we do not give details.
Lemma 7.2.
Suppose that a pronilpotent group G is generated by subgroups G1,…,Gt such that γi(G)=〈γi(G)∩Gj:1≤j≤t〉 for all i≥1. Then one has
G=G1⋯Gt.
As usual, for a profinite group G we denote by π(G) the set of prime divisors of the orders of finite continuous homomorphic images of G. We say that G is a π-group if π(G) is contained in π and G is a π′-group if π(G)∩π=∅. If π is a set of primes, we denote by Oπ(G) the maximal normal π-subgroup of G and by Oπ′(G) the maximal normal π′-subgroup.
We are ready to embark on the proof of Theorem 1.9.
Proof of Theorem 1.9 (2).
Let 𝒮 be the subset of all A-invariant open normal subgroups of G. For any N∈𝒮, set Q=G/N. Observe that, for any a∈A#, one has CQ(a)(d)=(CG(a))(d)N/N.
By hypothesis each x∈CG(a)(d) is Engel in G and this implies that all elements of CQ(a)(d) are Engel in Q. Hence Theorem 1.7 (2) tells us that Q(d) is Engel. By Zorn’s Theorem [10, Theorem III 6.3] Q(d) is nilpotent. Since G(d)≅lim←N∈𝒮(G/N)(d), we get that G(d) is a pronilpotent group. Thus, G(d) is the Cartesian product of its Sylow subgroups.
Choose a∈A#. For each positive integer i we set
Si={(x,y)∈G(d)×CG(a)(d):[x,iy]=1}.
Since the sets Si are closed in G(d)×CG(a)(d) and their union coincides with G(d)×CG(a)(d), by the Baire Category Theorem at least one Si has a non-empty interior. Therefore we can find an integer n, an open subgroup K in G(d), elements u∈G(d) and v∈CG(a)(d) such that
[ul,nvk]=1 for any l∈K and any k∈K∩CG(a)(d).
Let [G(d):K]=m and let π1=π(m) be the set of primes dividing m. Denote Oπ1′(G(d)) by T.
Since T is isomorphic to the image of K in G(d)/Oπ1(G(d)), it is easy to see that [T,nx]=1 for all x∈CT(a)(d). Thus, each element of CT(a)(d) is n-Engel in T.
The open subgroup K, the set π1 and the integer n depend only on the choice of a∈A#, so strictly speaking they should be denoted by Ka,πa and na, respectively. We choose such Ka,πa and na for any element a∈A#. Set π=⋃a∈A#πa, n=max{na:a∈A#} and R=Oπ′(G(d)). The choice of the set π guarantees that for each a∈A# every element of CR(a)(d) is n-Engel in R. Using a routine inverse limit argument, we deduce from Theorem
1.7 (2) that R is n1-Engel for some suitable integer n1. By [24, Theorem 5] we have that R is locally nilpotent. Let p1,p2,…,pr be the finitely many primes in π and let S1,…,Sr be the corresponding Sylow subgroups of G(d). Then G(d)=S1×⋯×Sr×R and therefore it is enough to show that each subgroup Si is locally nilpotent.
Let P be such a p-Sylow subgroup of G(d), for some p∈π. By the profinite version of Theorem 4.3 we have P=P1⋯Pt, where any Pj=P∩H and H is an A-special subgroup of G of degree d. Since 2d≤r-1, by the profinite version of Theorem 4.2 (2) each subgroup Pj⊆CG(a)(d), for some suitable a∈A#.
Choose arbitrary elements x1,…,xm in P. For i=1,2,…,m we write xi=xi1⋯xit, where xij belongs to Pj for j=1,2,…,t. Let Y be the subgroup generated by the orbits xijA. Put Yj=Y∩Pj. Since Y is generated by the orbits xijA and every Pj is an A-invariant subgroup, we deduce that Y is generated by Y1,…,Yt and, by Lemma 7.2, we have Y=Y1⋯Yt.
For our purpose it is sufficient to show that Y is nilpotent. To this end,
we denote by Dj=Dj(Y) the terms of the p-dimension central series of Y. Let L=Lp(Y) be the Lie algebra generated by D1/D2 associated with the pro-p group Y. Observe that D2 coincides with Φ(Y) and denote by V1,…,Vt the images of Y1,…,Yt in Y/Φ(Y). It follows that the Lie algebra L is generated by V1,…,Vt.
Since Y is A-invariant, the group A acts on L in the natural way.
Let A1,…,As be the distinct maximal subgroups of A.
In the same spirit of what was done in the proof of Proposition 5.1 we can define special subspaces of L of any weight and show that every special subspace W of L corresponds to a subgroup of an A-special subgroup of G of degree d.
Moreover, it follows from the profinite version of Proposition 4.2 (2) that every element of W corresponds to some element of CG(a)(d) for a suitable a∈A# and so, by Lemma 3.1, it is ad-nilpotent on L, since all elements of CG(a)(d) are Engel in G.
From the previous argument we deduce that L=〈V1,…,Vt〉 is generated by ad-nilpotent elements. As in the proof of Proposition 5.1 we extend the ground field 𝔽p by a primitive qth root of unity ω. Put L¯=L⊗𝔽p[ω] and identify, as usual, L with the 𝔽p-subalgebra L⊗1 of L¯.
In what follows we write X¯ to denote X⊗𝔽p[ω], for some subspace X of L.
Let W be a special subspace of L. We claim:
Claim 7.1.
Every element w of W¯ is ad-nilpotent in L¯.
Indeed, choose w∈W¯ and write
w=l0+ωl1+⋯+ωq-2lq-2,
for suitable elements l0,…,lq-2 of W that, in particular, correspond to some elements x0,…,xq-2 of CG(a)(d), for a suitable a∈A#.
Let denote by
K0=〈l0,ωl1,…,ωq-2lq-2〉
the subalgebra of L¯ generated by l0,ωl1,…,ωq-2lq-2.
Using now an argument analogous to that used in the proof of Proposition 5.1, we first apply Theorem 2.1 to show that K0 is nilpotent and, later, by appealing to Lemma 2.7, we conclude that w is ad-nilpotent in L¯, as claimed in Claim 7.1.
The group A acts on L¯ in the natural way and now the ground field is a splitting field for A. Since Y is finitely generated, we can choose finitely many elements v1,v2,… in V1¯∪⋯∪Vt¯, that generate the Lie algebra L¯, and each vi is a common eigenvector for A.
Let v be any Lie commutator in generators v1,v2… of L¯. Mimicking what we did in the proof of Proposition 5.1 and arguing by induction on the weight of v, we can show that v belongs to some W¯, where W is a special subspace of L. Thus, by Claim 7.1, v is ad-nilpotent.
This proves:
Claim 7.2.
Any commutator in v1,v2… is ad-nilpotent in L¯.
Furthermore, it follows from Lemma 7.1 (2) that L satisfies a multilinear Lie polynomial identity. The multilinear identity is also satisfied in L¯ and from Theorem 2.1 we deduce that L¯ is nilpotent. Since L embeds into L¯, we get that L is nilpotent as well.
According to Lazard [16] the nilpotency of L is equivalent to Y being p-adic analytic (for details see [16, A.1 in Appendice and Sections 3.1 and 3.4 in Chapter III] or [7, 1.(k) and 1.(o) in Interlude A]). By [7, 7.19 Theorem] Y admits a faithful linear representation over the field of p-adic numbers. A result of Gruenberg [8, Theorem 0] says that in a linear group the Hirsch–Plotkin radical coincides with the set of Engel elements.
Since Y=Y1⋯Yt is finitely generated and
each Yj is contained in CG(a)(d) for some suitable a∈A#, we deduce that Y is nilpotent.
The proof is complete.
∎
The proof of part (1) of Theorem 1.9 is analogous to that of item (2) and can be obtained by replacing every appeal to Theorem 1.7 (2), Theorem 4.3, Proposition 4.2 and Lemma
7.1 (2) in the proof of (2) by an appeal to Theorem 1.7 (1), Theorem 4.6, Proposition 4.5 and Lemma 7.1 (1), respectively. Therefore we omit the further details.
In what follows we give an outline of the proof of Theorem 1.10.
Proof of Theorem 1.10 (2).
With an argument similar to that used in the proof of Theorem 1.9 (2) and appealing to Theorem 1.8 (2) it easy to show that G(d) is pronilpotent and so it is the Cartesian product of its Sylow subgroups.
Choose a∈A#. Since CG(a)(d) is Engel, it follows from [24, Theorem 5] that CG(a)(d) is locally nilpotent. By [3, Lemma 2.5] there exist a positive integer n, elements u,v∈CG(a)(d) and an open subgroup H≤CG(a)(d) such that the law [x,ny]≡1 is satisfied on the cosets uH,vH. Let
[CG(a)(d):H]=m and π1=π(m). Denote Oπ1′(CG(a)(d)) by T. Thus, T satisfies the law [x,ny]≡1, that is, T is n-Engel.
By the result of Burns and Medvedev [6] the subgroup T has a nilpotent normal subgroup U such that T/U has finite exponent, say e. Set π2=π(e). The sets π1 and π2 depend on the choice of the element a∈A#, so strictly speaking they should be denoted by π1(a) and π2(a). For each such choice let πa=π1(a)∪π2(a). We repeat this argument for every element a∈A#. Set
π=⋃a∈A#πa and R=Oπ′(G(d)). Since all sets π1(a) and π2(a) are finite, so is π.
Let p1,…,pr be the finitely many primes in π and let S1,…,Sr be the corresponding Sylow subgroups of G(d). Then G(d)=S1×⋯×Sr×R.
The choice of the set π guarantees that CR(a)(d) is nilpotent for every a∈A#.
Using the routine inverse limit argument, we deduce from [2, Theorem 31] that R(d) is nilpotent. Thus R is solvable. We claim that R is an Engel group. Indeed, combining the profinite version of Proposition 4.2 (3) with Lemma 7.2, we obtain R=R1⋯Rt,
where Ri=R∩H and H is some A-special subgroup of G of degree d.
Choose arbitrary elements x,y∈R. It suffices to prove that 〈x,y〉 is nilpotent. Note that we can write x=x1⋯xt and y=y1⋯yt, where xi and yi belong to Ri, i=1,…,t. Consider Y=〈xiA,yiA:i≤t〉 and set K to be the abstract subgroup generated by the elements xiA,yiA. Since K is a dense subgroup of Y, in order to prove that Y is nilpotent it is enough to prove that K is nilpotent.
By construction and since 2d≤r-2, there exists an element a∈A# such that the centralizer CG(Aj) is contained in CG(a) and each subgroup Ri is contained in CG(a)(d). Thus:
Claim 7.3.
Each element x∈Ri is Engel in CG(Aj), for any j≤s.
In the same spirit of what was done in the proof of Theorem 1.8 (2), it is possible to show, by induction on the derived length of K, that all generators of K are Engel elements in K. Now by a well-known result of Gruenberg [19, Theorem 12.3.3] we conclude that K is nilpotent and, so, Y is nilpotent, as well.
In particular, we deduce that R is Engel, as desired. Thus, by [24, Theorem 5] we have that R is locally nilpotent.
Since G(d)=S1×⋯×Sr×R, for our purpose it is sufficient to prove that each subgroup Si is locally nilpotent, for i=1,…,r.
Let P be such a p-Sylow subgroup of G(d), for some p∈π. By the profinite version of Theorem 4.3 we have P=P1⋯Pt, where any Pj=P∩H and H is an A-special subgroup of G with degree d.
Choose arbitrary elements x1,…,xm in P. Let us write xi=xi1⋯xit for i=1,…,m, where each xij belongs to Pj and so to CG(a)(d) for a suitable a∈A#. Let X be the subgroup generated by the orbits xijA and let L=Lp(X). By the assumptions we have L=∑j≤sCL(Aj) and by using the Lie theoretical machinery it is possible to prove that L is nilpotent.
According to Lazard [16] the nilpotency of L is equivalent to X being p-adic analytic. The Lubotzky and Mann theory [17, 18] now tells us that X is of finite rank, that is, all closed subgroups of X are finitely generated. In particular, we conclude that CX(a)(d) is finitely generated for every a∈A#. From [24, Theorem 5] it follows that CX(a)(d) is nilpotent, and by the profinite quantitative version of [2, Theorem 31], X(d) is nilpotent. Then X is soluble.
Finally, by mimicking what we did above for Y, we can prove that X is nilpotent. This concludes the proof.
∎
The proofs of part (1) and (2) of Theorem 1.10 are very similar. We conclude noting that the proof of item (1) of Theorem 1.10 can be obtained by replacing every appeal to Theorem 1.8 (2), Theorem 4.3, Proposition 4.2 and Lemma 7.1 (2) in the proof of (2) by an appeal to Theorem 1.8 (1) Theorem 4.6, Proposition 4.5 and Lemma 7.1 (1), respectively. Therefore we will omit further details.
Acknowledgements
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and
Danilo Sanção da Silveira
