arXiv:1707.06889v1 [] 20 Jul 2017

Profinite groups and centralizers of coprime
automorphisms whose elements are Engel
Cristina Acciarri and Danilo Sanção da Silveira
Abstract. Let q be a prime, n a positive integer and A an elementary
abelian group of order q r with r ≥ 2 acting on a finite q ′ -group G. The
following results are proved.
We show that 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, and 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.
Assuming r ≥ 3 we prove that 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, and 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.
Analogue (non-quantitative) results for profinite groups are also
obtained.

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.
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. Remind 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 next
theorem was obtained in [14].
2010 Mathematics Subject Classification. 20D45, 20E18, 20F40, 20F45.
Key words and phrases. Profinite groups, Automorphisms, Centralizers, Engel
elements.
This work was supported by the Conselho Nacional de Desenvolvimento Cientı́fico e
Tecnológico (CNPq), Brazil.
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Theorem 1.1. Let q be a prime, n a positive integer and A an elementary abelian group of order q 2 . 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 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 [25, 26] are
combined with the Lubotzky–Mann theory of powerful p-groups [17], and
a theorem of Bahturin and Zaicev on Lie algebras admitting a group of
automorphisms whose fixed-point subalgebra is PI [5].
A profinite (non-quantitative) version of the above theorem was established in [21]. In the context of profinite groups all the usual concepts of
groups 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 G coprimely 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 q 2 . 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 [22] 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,n y] are defined inductively by the rule
[x,0 y] = x,

[x,n y] = [[x,n−1 y], 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,n x] = 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,n x] = 1. The group G is n-Engel if all elements of
G are n-Engel. The following result was proved in [22].
Theorem 1.3. Let q be a prime, n a positive integer and A an elementary abelian group of order q 2 . 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 q 2 . Suppose that A acts coprimely on a profinite group G and assume

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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 [23, 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 q 3 . 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 q 3 . 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, noted above between Theorem 1.3 and Theorem 1.4, 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 q r with r ≥ 2 acting on a finite q ′ -group
G.
(1) 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.
(2) 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 q r with r ≥ 3 acting on a finite q ′ -group
G.

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(1) 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.
(2) 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 and

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 q r with r ≥ 2 acting coprimely on a profinite
group G.
(1) If all elements in γr−1 (CG (a)) are Engel in G for any a ∈ A# , then
γr−1 (G) is locally nilpotent.
(2) 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 q r with r ≥ 3 acting coprimely on a profinite
group G.
(1) If all elements in γr−2 (CG (a)) are Engel in CG (a) for any a ∈ A# ,
then γr−2 (G) is locally nilpotent.
(2) 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 [19, 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.

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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,0 y] = x; [x,m y] = [[x,m−1 y], y], for all positve 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,n a] = 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 [25]. A detailed proof was published in [27].
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 for soluble
groups A [5] and later extended by Linchenko to the general case [16],
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 [20]).
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 L is
nilpotent of {f, K, m, n}-bounded class.
Theorem 2.4. Let L be as in Theorem 2.2 and 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 characterisitic 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.

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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 [22] it was established the following result 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 ] = 0.
| {z }
u

Recall that the identity
X

[y, xσ(1) , . . . , xσ(n) ] ≡ 0

σ∈Sn

is called linearized n-Engel identity. In general, 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 [22] for further details.
Theorem 2.8. Let F be the free Lie ring and f an element of F (Lie
polynomial) such that f 6∈ 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, 20]). 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 L∗i = Gi /Gi+1 , written

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additively. Commutation in G induces a binary operation [, ] in L∗ (G). For
homogeneous elements xGi+1 ∈ L∗i , yGj+1 ∈ L∗j the operation is defined by
[xGi+1 , yGj+1 ] = [x, y]Gi+j+1 ∈ L∗i+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 Gpi ≤ 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 of 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 [20] 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
M (Gj ∩ H)Gj+1
L∗ (G, H) =
Gj+1
j≥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. [23, 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, n g] = 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 finte p-group and suppose that
Lp (G) is nilpotent of class c. Then P has a powerful characteristic subgroup
of {p, c, d}-bounded index.

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Remind that powerful p-groups were introduced by Lubotzky and Mann
in [17]. A finite p-group P is said to be powerful if and only if [P, P ] ≤ P p
for p 6= 2 (or [P, P ] ≤ P 4 for p = 2), where P i 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
P i 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
[P n1 , . . . , P ns ] ≤ 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 informations on that topic. Let
us start by recalling the definition.
Definition 4.1. Let A be 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.
(1) 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.
(2)Suppose that k ≥ 1 and the A-special subgroups of G of degree k − 1
are defined. Then H is an A-special subgroup of G of degree k if and only
if there exist A-special subgroups J1 , 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 q r , then for 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 q r
with r ≥ 2 acting on a finite q ′ -group G and A1 , . . . , As be the maximal
subgroups of A. Let k ≥ 0 be an integer.
(1) 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.

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(2) 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
k
such that |A/B| ≤ q 2 .
(3) Let Rk be the subgroup generated by all A-special subgroups of G of
degree k. Then Rk = G(k) .
(4) 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 q r 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 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. Let A be 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.
(1) 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.
(2) Suppose that k ≥ 2 and the γ-A-special subgroups of G of degree
k − 1 are defined. Then H is 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 q r
with r ≥ 2 acting on a finite q ′ -group G and A1 , . . . , As the maximal subgroups of A. Let k ≥ 1 be an integer.
(1) 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.
(2) If k ≤ r − 1 and H is a γ-A-special subgroup of G of degree k, then
H ≤ γk (CG (B)) for some subgroup B ≤ A such that |A/B| ≤ q k .
(3) Let Rk be the subgroup generated by all γ-A-special subgroups of G
of degree k. Then Rk = γk (G).
(4) 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 .

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Theorem 4.6. Let A be an elementary abelian q-group of order q r with
r ≥ 2 acting on a finite q ′ -group G. Let p be a prime and P 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, K be subgroups of a profinite group G. Remind 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.
(1) 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.
(2) Suppose that k ≥ 1 and the A-special subgroups of G of degree k − 1
are defined. Then H is an A-special subgroup of G of degree k if and only
if there exist A-special subgroups J1 , 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
Theorem 1.7(2). 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:

PROFINITE GROUPS AND CENTRALIZERS OF COPRIME AUTOMORPHISMS

11

(1) Lp (H) is nilpotent of {n, p, q, r, t}-bounded class.
(2) There exist positive integers e, 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 , 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 , W2 correspond respectively to some J1 , 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,
(1)

every element of W is ad-nilpotent of index at most n,

since all elements of CG (a)(d) are n-Engel in G.
From the previous argument we deduce that L = hV1 , . . . , Vt i 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 that
(2)

there exists a {n, q}-bounded number u such that every
element w of W is ad-nilpotent of index at most u.

Indeed, chose w ∈ W and write
w = l0 + ωl1 + · · · + ω q−2 lq−2 ,

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C. ACCIARRI AND D. S. DA SILVEIRA

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# . Let denote by K =
hl0 , ωl1 , . . . , ω q−2 lq−2 i the subring of L generated by l0 , ωl1 , . . . , ω q−2 lq−2 and
put H0 = hx0 , . . . , xq−2 i. 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 , . . . , ω q−2 lq−2 is of the form ω α v
for some v ∈ W , and so, by (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 a positive integer u, depending only on n and q,
such that [L,u K] = 0. Since w ∈ K, we conclude that w is ad-nilpotent in
L with {n, q}-bounded index, as claimed in (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 = ω λ x. 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 v has weight at least 2. Write v = [w1 , w2 ] for some w1 ∈ W1 and
w2 ∈ W2 , where W1 , 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 (2), v is ad-nilpotent of index at most u. This proves that
(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,n δ2d (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, 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 the items (1) and (2) of the previous proposition
are very similar. As for the proof of item (1), we only observe that it can

PROFINITE GROUPS AND CENTRALIZERS OF COPRIME AUTOMORPHISMS

13

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 Aspecial subgroups of G of degree d and Proposition 4.2(2) tells us that any Aspecial subgroup of G of degree d is contained in CG (B)(d) for some suitable
nontrivial subgroup B ≤ A such that |A/B| ≤ q r−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, 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, we have
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 arbitrarily x, y ∈ P . Let us write x = x1 · · · xt and y = y1 · · · yt ,
where xi and yi belong to Pi . In what follows we will show that hx, yi is
nilpotent of {n, q, r}-bounded class. Let Y be the subgroup generated by
A
the orbits xA
i and yi , for i = 1, . . . , t. Note that Y is generated by a {q, r}bounded number of elements. Since the subgroup hx, yi is contained in Y ,
it is enough to show that Y is nilpotent of {n, q, r}-bounded class.
Set Yi = Pi ∩ Y , for i = 1, . . . , t and note that every Yi is a subgroup of
A
CG (a)(d) for a suitable a ∈ A# . Since Y = hxA
i , yi | i = 1, . . . , ti and every
Pi is an A-invariant subgroup we have Y = hY1 , . . . , Yt i. 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, 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
hx, yi. 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 tell us
that Y has a powerful characteristic subgroup K of {n, q, r}-bounded index.
It follows from [18, 6.1.8(ii), p. 164] that K has a {n, q, r}-bounded rank.
Put R = K e and assume that R 6= 1. Note that, if p 6= 2, then we have
2

2

[R, R] ≤ [K, K]e ≤ K pe = Rpe ,
and if p = 2, then we have
[R, R] ≤ R4e .

14

C. ACCIARRI AND D. S. DA SILVEIRA

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 6= 2, that
γc (R)e ≤ γc+1 (R) = [R′ ,c−1 R] ≤ [Rpe ,c−1 R] ≤ γc (R)pe
and, if p = 2, that
γc (R)e ≤ γc (R)4e .
Hence γc (R)e = 1. Since γc (R) is also powerful and generated by a {n, q, r}bounded number of elements, we conclude that γc (R) is of {n, q, r}-bounded
order, since it is a product of a {n, q, r}-bounded number of cyclic subgroups.
It follows that R has a {n, q, r}-bounded derived length. Remind that R =
K e 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 [22, Lemma 4.1] tell us that Y has {n, q, r}-bounded nilpotency
class, and the same holds for hx, yi, 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
Theorem 1.7(1). 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:
(1) Lp (H) is nilpotent of {n, p, q, r, t}-bounded class.
(2) There exist positive integers e, 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 the 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
Theorem 1.8(2). 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 ,

PROFINITE GROUPS AND CENTRALIZERS OF COPRIME AUTOMORPHISMS

15

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:
(1) Lp (H) is nilpotent of {n, p, q, r, t}-bounded class.
(2) There exist positive integers e, 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
d
of A such that |A/B| ≤ q 2 . 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
that
(4)

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 we have L =
P
j≤s CL (Aj ). Now taking into account that every special subspace W of L is
contained in some L(H, Hi ) and that, by (4), we have [CL (Aj ),n L(H, Hi )] =
0, 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 the part (2) of Theorem 1.8.
Proof. By 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) ,
d
for some subgroup B of A such that |A/B| ≤ q 2 .
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,
(5)

each element l ∈ Pi is n-Engel in CG (Aj ), for any j ≤ s.

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C. ACCIARRI AND D. S. DA SILVEIRA

Choose arbitrarily x, y ∈ P . We will show that the subgroup hx, yi 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
A
subgroup Y = xA
i , yi |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 [18, 6.1.8(ii),
p. 164], of {n, q, r}-bounded rank as well.
Thus each CN (a)(d) is an n-Engel subgroup and can be generated by
a {n, q, r}-bounded number of elements. Zelmanov noted in [24] 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] tell us that N (d)
is nilpotent of {n, q, r}-bounded class. This implies that Y has {n, q, r}bounded derived length l, say.
In view of [22, 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] ⊆ M.
| {z }
c1
P
Since M is abelian and A-invariant we can write M = j≤s CM (Aj ). By (5)
we have [M, x, . . . , x] = 1. Thus, [Y, x, . . . , x] = 1. This implies that hx, yi
| {z }
| {z }
n

c1 +n

is nilpotent of {n, q, r}-bound class, as desired.
By the argument above we get 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
Theorem 1.8(1). 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:
(1) Lp (H) is nilpotent of {n, p, q, r, t}-bounded class.
(2) There exist positive integers e, c depending only n, q, r and t, such
that eγc (L(H)) = 0.

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17

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 g1 H, g2 H, . . . , gk H if w(g1 h1 , g2 h2 , . . . , gk hk ) = 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:
(1) 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.
(2) 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 the 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). For each integer i we define the set
{z
}
|
2k +1

Si = {(t, t1 , . . . , t2k ) ∈ T : [t,i δk (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 nonempty interior. Therefore, we can find an open subgroup H in CG (A),
elements g, g1 , . . . , g2k ∈ CG (A) and an integer n such that the identity
[x,n δk (y1 , . . . , y2k )] ≡ 1 is satisfied on the cosets gH, g1 H, . . . , g2k H. Thus,
the Wilson-Zelmanov result [23, Theorem 1] tell 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) = hγi (G) ∩ Gj : 1 ≤ j ≤ ti for all i ≥ 1.
Then 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.

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C. ACCIARRI AND D. S. DA SILVEIRA

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 CQ (a)(d) = (CG (a))(d) N/N , for any a ∈ A# . 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) tell us that Q(d) is Engel. By Zorn’s Theorem
(d)
[10, Theorem III 6.3] Q(d) is nilpotent. Since G(d) ∼
= lim
←−N ∈S (G/N ) , 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,i y] = 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,n vk] = 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,n x] = 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 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 [23,
Theorem 5] R is locally nilpotent. Let p1 , p2 , . . . , pr be the finitely many
primes in π and 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 xA
ij . Put Yj = Y ∩ Pj . Since Y is generated
A
by the orbits xij 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. We denote by
Dj = Dj (Y ) the terms of the p-dimension central series of Y . Let L = Lp (Y )

PROFINITE GROUPS AND CENTRALIZERS OF COPRIME AUTOMORPHISMS

19

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 = hV1 , . . . , Vt i 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 that
(6)

every element w of W is ad-nilpotent in L.

Indeed, choose w ∈ W and write
w = l0 + ωl1 + · · · + ω q−2 lq−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 =
hl0 , ωl1 , . . . , ω q−2 lq−2 i the subalgebra of L generated by l0 , ωl1 , . . . , ω q−2 lq−2 .
Using a analogous argument 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 (6).
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 (6), v is adnilpotent. This proves that
(7)

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.

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C. ACCIARRI AND D. S. DA SILVEIRA

According to Lazard [15] the nilpotency of L is equivalent to Y being
p-adic analytic (for details see [15, A.1 in Appendice and Sections 3.1 and
3.4 in Ch. 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 HirschPlotkin 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, [23, Theorem 5] guarantees
that CG (a)(d) is locally nilpotent. By [3, Lemma 2.5] there exists a positive
integer n, elements u, v ∈ CG (a)(d) and an open subgroup H ≤ CG (a)(d) such
that the law [x,n y] ≡ 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,n y] ≡ 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 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 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 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 that R = R1 · · · Rt , where Ri = R ∩ H
and H is some A-special subgroup of G of degree d. Choose arbitrarily
x, y ∈ R. Therefore it is sufficient to prove that hx, yi is nilpotent. Note
that we can write x = x1 · · · xt and y = y1 · · · yt , where each xi and yi
A
belongs to Ri , for i = 1, . . . , t. Consider Y = xA
i , yi : i ≤ t and set K be
A
the abstract subgroup generated by the elements xA
i and yi . Since K is

PROFINITE GROUPS AND CENTRALIZERS OF COPRIME AUTOMORPHISMS

21

a dense subgroup of Y , in order to prove that Y is nilpotent is enough to
prove that K is nilpotent.
By construction and since 2d ≤ r − 2, there exists a ∈ A# such that the
centralizer CG (Aj ) is contained in CG (a) and each subgroup Ri is contained
in CG (a)(d) . Thus
(8)

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
[18, 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 [23, Theorem
5] 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 xA
ij and let
P
L = Lp (X). By the assumptions we have L = j≤s CL (Aj ) and using the
Lie theoretical machinery it is possible to prove that L is nilpotent.
According to Lazard [15] the nilpotency of L is equivalent to X being
p-adic analytic. The Lubotzky and Mann theory [17] 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# .
It follows from [23, Theorem 5] that CX (a)(d) is nilpotent. 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.
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23

Department of Mathematics, University of Brasilia, 70910-900 Brası́lia
DF, Brazil
E-mail address: acciarricristina@yahoo.it
Department of Mathematics, Federal University of Goias, 75704-020 Catalão
GO, Brazil
E-mail address: sancaodanilo@ufg.br

