arXiv:1601.05380v1 [] 20 Jan 2016

THE NON-ABELIAN TENSOR SQUARE OF
RESIDUALLY FINITE GROUPS
R. BASTOS AND N. R. ROCCO

Abstract. Let m, n be positive integers and p a prime. We denote by ν(G) an extension of the non-abelian tensor square G ⊗ G
by G × G. We prove that if G is a residually finite group satisfying
some non-trivial identity f ≡ 1 and for every x, y ∈ G there exists a p-power q = q(x, y) such that [x, y ϕ ]q = 1, then the derived
subgroup ν(G)′ is locally finite (Theorem A). Moreover, we show
that if G is a residually finite group in which for every x, y ∈ G
there exists a p-power q = q(x, y) dividing pm such that [x, y ϕ ]q is
left n-Engel, then the non-abelian tensor square G ⊗ G is locally
virtually nilpotent (Theorem B).

1. Introduction
The non-abelian tensor square G ⊗ G of a group G as introduced by
Brown and Loday [BL84, BL87] is defined to be the group generated
by all symbols g ⊗ h, g, h ∈ G, subject to the relations
gg1 ⊗ h = (g g1 ⊗ hg1 )(g1 ⊗ h) and g ⊗ hh1 = (g ⊗ h1 )(g h1 ⊗ hh1 )
for all g, g1, h, h1 ∈ G.
This is a particular case of a non-abelian tensor product G ⊗ H, of
groups G and H, under the assumption that G and H acts one on each
other and on itself by conjugation.
We observe that the defining relations of the tensor square can be
viewed as abstractions of commutator relations; thus in [Roc91] the
following related construction is considered: Let G and H be groups
and ϕ : H → H ϕ an isomorphism (H ϕ is an isomorphic copy of H,
where h 7→ hϕ , for all h ∈ H). Define the group ν(G) to be
ϕ

ν(G) := hG, Gϕ | [g1 , g2 ϕ ]g3 = [g1 g3 , (g2g3 )ϕ ] = [g1 , g2 ϕ ]g3 , gi ∈ Gi.
The motivation for studying ν(G) is the commutator connection:
Proposition 1.1. ([Roc91, Proposition 2.6]) The map Φ : G ⊗ G →
[G, Gϕ ], defined by g ⊗ h 7→ [g, hϕ ], ∀g, h ∈ G, is an isomorphism.
2010 Mathematics Subject Classification. 20F45, 20E26, 20F40.
Key words and phrases. Residually finite groups; Engel elements.
1

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BASTOS AND ROCCO

From now on we identify the tensor square G ⊗ G with the subgroup
[G, Gϕ ] of ν(G) (see for instance [NR12] for more details).
It is a natural task to study structural aspects of ν(G) and G ⊗ G for
different classes of infinite groups G. Moravec [Mor08] proves that if G
is locally finite then so is G ⊗ G (and, consequently, also ν(G)). In the
article [BR] the authors establish some structural results concerning
ν(G) when G is a finite-by-nilpotent group. In the present paper we
are concerned with the non-abelian tensor square of residually finite
groups.
According to the solution of the Restricted Burnside Problem (Zelmanov, [Zel91a, Zel91b]) every residually finite group of finite exponent is locally finite. Another interesting result in this context, due to
Shumyatsky [Shu05] states that if G is a residually finite group satisfying a non-trivial identity and generated by a normal commutator-closed
set of p-elements, then G is locally finite. A subset X of a group G is
called commutator-closed if [x, y] ∈ X whenever x, y ∈ X. In a certain
way, our results can be viewed as generalizations of some of the above
results.
Theorem A. Let p a prime. Let m be a positive integer and G a residually finite group satisfying some non-trivial identity f ≡ 1. Suppose
that for every x, y ∈ G there exists a positive integer q = q(x, y) dividing pm such that the tensor [x, y ϕ ]q = 1. Then the derived subgroup
ν(G)′ is locally finite.
Let G be a group. For elements x, y of G we define [x, 1 y] = [x, y] and
[x, i+1 y] = [[x, i y], y] for i ≥ 1, where [x, y] = x−1 y −1 xy. An element
y ∈ G is called left n-Engel if for any x ∈ G we have [x, n y] = 1. The
group G is called a left n-Engel group if [x, n y] = 1 for all x, y ∈ G.
Another result that was deduced following the solution of the Restricted Burnside Problem is that any residually finite n-Engel group
is locally nilpotent (J. S. Wilson, [Wil91]). Later, Shumyatsky proved
that if G is a residually finite group in which every commutator is
left n-Engel in G, then the derived subgroup G′ is locally nilpotent
[Shu99b, Shu00]. In the present paper we establish the following related result.
Theorem B. Let m, n be positive integers and p a prime. Suppose that
G is a residually finite group in which for every x, y ∈ G there exists
a positive integer q = q(x, y) dividing pm such that the element [x, y ϕ ]q
is left n-Engel in ν(G). Then the non-abelian tensor square G ⊗ G is
locally virtually nilpotent.

NON-ABELIAN TENSOR SQUARE

3

A natural question arising in the context of Theorem B is whether
the theorem remains valid when q is allowed to be an arbitrary positive
integer, rather than a p-power (see Proposition 4.4 in Section 4 for
details).
The paper is organized as follows. Section 2 presents terminology
and some preparatory results that are later used in the proofs of our
main results. In Section 3 we describe some important ingredients of
what is often called Lie methods in group theory. These are essential
in the proof of Theorem B. Section 4 contains the proofs of the main
theorems.
2. Preliminary results
The following basic properties are consequences of the defining relations of ν(G) and the commutator rules (see [Roc91, Section 2] for
more details).
Lemma 2.1. The following relations hold in ν(G), for all g, h, x, y ∈ G.
ϕ
(i) [g, hϕ ][x,y ] = [g, hϕ ][x,y] ;
(ii) [g, hϕ , xϕ ] = [g, h, xϕ ] = [g, hϕ , x] = [g ϕ , h, xϕ ] = [g ϕ , hϕ , x] =
[g ϕ , h, x];
(iii) If h ∈ G′ (or if g ∈ G′ ) then [g, hϕ ][h, g ϕ ] = 1;
(iv) [g, [h, x]ϕ ] = [[h, x], g ϕ ]−1 ;
(v) [[g, hϕ ], [x, y ϕ ]] = [[g, h], [x, y]ϕ ].
Lemma 2.2. Let G be a group. Then X = {[a, bϕ ] | a, b ∈ G} is a
normal commutator-closed subset of ν(G).
ϕ

Proof. By definition, [a, bϕ ]y = [a, bϕ ]y = [ay , (by )ϕ ], for every a, b, y ∈
G. It follows that X is a normal subset of ν(G). Now, it remains to
prove that X is a commutator-closed subset of ν(G). Choose arbitrarily
elements a, b, c, d ∈ G and consider the elements [a, bϕ ], [c, dϕ ] ∈ X. By
Lemma 2.1 (v),
[[a, bϕ ], [c, dϕ ]] = [[a, b], [c, d]ϕ ] ∈ X.
The result follows.



The epimorphism ρ : ν(G) → G, given by g 7→ g, hϕ 7→ h, induces
the derived map
ρ′ : [G, Gϕ ] → G′ ,
[g, hϕ ] 7→ [g, h] for all g, h ∈ G. In the notation of [Roc94, Section 2],
let µ(G) denote the kernel of ρ′ . In particular,
[G, Gϕ ] ∼ ′
=G.
µ(G)

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The next lemma is a particular case of Theorem 3.3 in [Roc91].
Lemma 2.3. Let G be a group. Then the derived subgroup
ν(G)′ = ([G, Gϕ ] · G′ ) · (G′ )ϕ ,
where “·” denotes an internal semi-direct product.
Lemma 2.4. Let m, n be positive integers and x, y be elements in a
group G. If [x, y ϕ ]m is left n-Engel element in ν(G), then the power
[x, y]m is left n-Engel in G.
Proof. By assumption, [z,n [x, y ϕ ]m ] = 1 for every z ∈ G. It follows
that
1 = ρ([z,n [x, y ϕ ]m ]) = [z,n [x, y]m ],
for every z ∈ G, i.e., [x, y]m is left n-Engel in G, and the lemma follows.

The remainder of this section will be devoted to describe certain
finiteness conditions for residually finite and locally graded groups. Recall that a group is locally graded if every non-trivial finitely generated
subgroup has a proper subgroup of finite index. Interesting classes of
groups (e.g. locally finite groups, locally nilpotent groups, residually
finite groups) are locally graded.
We need the following result, due to Shumyatsky [Shu05].
Theorem 2.5. Let G be a residually finite group satisfying some nontrivial identity f ≡ 1. Suppose G is generated by a normal commutatorclosed set X of p-elements. Then G is locally finite.
Next, we extend this result to the class of locally graded groups.
Lemma 2.6. Let p be a prime. Let G be a locally graded group satisfying
some non-trivial identity f ≡ 1. Suppose that G is generated by finitely
many elements of finite order and for every x, y ∈ G there exists a ppower q = q(x, y) such that [x, y]q = 1. Then G is finite.
Proof. Let R be the finite residual of G, i.e., the intersection of all
subgroups of finite index in G. If R = 1, then G is residually finite.
Since the set of all commutators is a normal commutator-closed subset
of p-elements, we have G′ is locally finite (Theorem 2.5). Moreover,
G/G′ is generated by finitely many elements of finite order. It follows
that G/G′ is finite and so G′ is finitely generated. Consequently, G is
finite. So, we can assume that R 6= 1. By the same argument, G/R is
finite and thus R is finitely generated. As R is locally graded we have
that R contains a proper subgroup of finite index in G, which gives a
contradiction.


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It is easy to see that a quotient of a residually finite group need
not be residually finite (see for instance [Rob96, 6.1.9]). This includes
the class of locally graded groups. However, the next result gives a
sufficient condition for a quotient to be locally graded.
Lemma 2.7. (Longobardi, Maj, Smith, [LMS95]) Let G be a locally
graded group and N a normal locally nilpotent subgroup of G. Then
G/N is locally graded.
3. On Lie Algebras Associated with Groups
Let L be a Lie algebra over a field K. We use the left normed
notation: thus if l1 , l2 , . . . , ln are elements of L, then
[l1 , l2 , . . . , ln ] = [. . . [[l1 , l2 ], l3 ], . . . , ln ].
We recall that an element a ∈ L is called ad-nilpotent if there exists a
positive integer n such that [x, n a] = 0 for all x ∈ L. When n is the
least integer with the above property then we say that a is ad-nilpotent
of index n.
Let X ⊆ L be any subset of L. By a commutator of elements in
X, we mean any element of L that could be obtained from elements of
X by means of repeated operation of commutation with an arbitrary
system of brackets including the elements of X. Denote by F the
free Lie algebra over K on countably many free generators x1 , x2 , . . . .
Let f = f (x1 , x2 , . . . , xn ) be a non-zero element of F . The algebra
L is said to satisfy the identity f = 0 if f (l1 , l2 , . . . , ln ) = 0 for any
l1 , l2 , . . . , ln ∈ L. In this case we say that L is PI. Now, we recall
an important theorem of Zelmanov [Zel90, Theorem 3] that has many
applications in roup Theory.
Theorem 3.1. Let L be a Lie algebra generated by l1 , l2 , . . . , lm . Assume that L is PI and that each commutator in the generators is adnilpotent. Then L is nilpotent.
Let G be a group and p a prime. In what follows,
Y
k
Di = Di (G) =
(γj (G))p
jpk >i

denotes the i-th dimension subgroup of G in characteristic p. These
subgroups form a central series of G known as the Zassenhaus-JenningsLazard seriesL(this can be found in Shumyatsky [Shu00, Section 2]).
Set L(G) =
Di /Di+1 . Then L(G) can naturally be viewed as a Lie
algebra over the field Fp with p elements. The subalgebra of L(G) generated by D1 /D2 will be denoted by Lp (G). The nilpotency of Lp (G)

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BASTOS AND ROCCO

has strong influence in the structure of a finitely generated group G.
The following result is due to Lazard [Laz65].
Theorem 3.2. Let G be a finitely generated pro-p group. If Lp (G) is
nilpotent, then G is p-adic analytic.
Let x ∈ G, and let i = i(x) be the largest integer such that x ∈ Di .
We denote by x̃ the element xDi+1 ∈ L(G). Now, we can state one
condition for x̃ to be ad-nilpotent.
Lemma 3.3. (Lazard, [Laz54, page 131]) For any x ∈ G we have
(ad x̃)q = ad (xeq ).

Remark 3.4. We note that q in Lemma 3.3 does not need to be a ppower. In fact, is easy to see that if ps is the maximal p-power dividing
q, then x̃ is ad-nilpotent of index at most ps .
The following result is an immediate corollary of [WZ92, Theorem
1].
Lemma 3.5. Let G be any group satisfying a group-law. Then L(G) is
P I.
For a deeper discussion of applications of Lie methods to group theory we refer the reader to [Shu00].
4. Proof of the main results
By a well-known theorem of Schmidt [Rob96, 14.3.1], the class of
locally finite groups is closed with respect to extension. By Lemma
2.3, ν(G)′ = ([G, Gϕ ] · G′ ) · (G′ )ϕ . From this and Proposition 1.1 we
conclude that the non-abelian tensor square G ⊗ G is locally finite if
and only if ν(G)′ is locally finite. The next result will be needed in the
proof of Theorem A.
Proposition 4.1. Let G be a group in which G′ is locally finite. The
following properties are equivalents.
(a) [G, Gϕ ] is locally finite;
(b) ν(G)′ is locally finite;
(c) For every x, y ∈ G there exists a positive integer m = m(x, y)
such that [x, y ϕ ]m = 1.
Proof. (a) ⇒ (b): It is sufficient to see that ν(G)′ = ([G, Gϕ ]·G′)·(G′)ϕ
(Theorem 2.3).
(b) ⇒ (c): Since ν(G)′ is locally finite, it follows that every element
in ν(G)′ has finite order. In particular, for every x, y ∈ G the element

NON-ABELIAN TENSOR SQUARE

7

[x, y ϕ ] has finite order.
(c) ⇒ (a): Let us first prove that [G, Gϕ ] is locally finite. Let W
be a finitely generated subgroup of [G, Gϕ ]. Since the factor group
[G, Gϕ ]/µ(G) is isomorphic to G′ , it follows that W is central-by-finite.
By Schur’s Lemma [Rob96, 10.1.4], the derived subgroup W ′ is finite.
Since W/W ′ is an abelian group generated by finite many elements of
finite order, it follows that W/W ′ is also finite. Thus [G, Gϕ ] is locally
finite. Now, Lemma 2.3 shows that ν(G)′ = ([G, Gϕ ] · G′ ) · (G′ )ϕ .
Therefore ν(G)′ is locally finite, by [Rob96, 14.3.1].

We are now in a position to prove Theorem A.
Theorem A. Let p be a prime and G a residually finite group satisfying
some non-trivial identity f ≡ 1. Suppose that for every x, y ∈ G there
exists a p-power q = q(x, y) such that [x, y ϕ ]q = 1. Then the derived
subgroup ν(G)′ is locally finite.
Proof. The proof is completed by showing that G′ is locally finite
(Proposition 4.1). Set
X = { [a, bϕ ]; a, b ∈ G}.
By Lemma 2.2, X is a normal commutator-closed subset of p-elements
in ν(G). Since [G, Gϕ ]/µ(G) ∼
= G′ , we conclude that the derived subgroup G′ is residually finite satisfying the identity f ≡ 1. In the same
manner we can see that for every x, y there exists a p-power q = q(x, y)
such that [x, y]q = 1. Theorem 2.5 now implies shows that G′ is locally
finite. The proof is complete.

Remark 4.2. For more details concerning residually finite groups in
which the derived subgroup is locally finite see [Shu99a, Shu05].
Now we will deal with Theorem B: Let m, n be positive integers and p
a prime. Suppose that G is a residually finite group in which for every
x, y ∈ G there exists a p-power q = q(x, y) dividing pm such that [x, y ϕ ]q
is left n-Engel in ν(G). Then the non-abelian tensor square G ⊗ G is
locally virtually nilpotent.
We denote by N the class of all finite nilpotent groups. The following
result is a straightforward corollary of [Wil91, Lemma 2.1].
Lemma 4.3. Let G be a finitely generated residually-N group. For each
prime p, let Rp be the intersection of all normal subgroups of G of finite
p-power index. If G/Rp is nilpotent for each p, then G is nilpotent.
The next result will be helpful in the proof of Theorem B.

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BASTOS AND ROCCO

Proposition 4.4. Let m, n be positive integers and p a prime. Suppose
that G is a residually finite group in which for every x, y ∈ G there
exists a p-power q = q(x, y) dividing pm such that [x, y]q is left n-Engel.
m
Then (G′ )p is locally nilpotent. Moreover, the derived subgroup G′ is
locally virtually nilpotent.
Proof. Let K be the derived subgroup of G. Choose an arbitrary commutator x ∈ G and let q = q(x) be a positive integer q = q(x) dividing pm such that xq is left n-Engel. It suffices to prove that the
normal closure of xq in G, h(xq )h | h ∈ Gi, is locally nilpotent. In
particular, xq ∈ HP (G). Let b1 , . . . , bt be finitely many elements in G.
Let hi = (xq )bi , i = 1, . . . , t and H = hh1 , . . . , ht i. We only need to
show that H is soluble (Gruenberg’s Theorem [Rob96, 12.3.3]). As a
consequence of Lemma 4.3, we can assume that H is residually-p for
some prime p. Let L = Lp (H) be the Lie algebra associated with the
Zassenhaus-Jennings-Lazard series
H = D1 ≥ D2 ≥ · · ·
of H. Then L is generated by h̃i = hi D2 , i = 1, 2, . . . , t. Let h̃ be any
Lie-commutator in h̃i and h be the group-commutator in hi having the
same system of brackets as h̃. Since for any group commutator h in
h1 . . . , ht there exists a positive integer q dividing pm such that hq is
left n-Engel, Lemma 3.3 shows that any Lie commutator in h̃1 . . . , h̃t
is ad-nilpotent. On the other hand, for every commutator x = [a1 , b1 ]
there exists a positive integer q = q(x) dividing pm such that xq is left
n-Engel, then G satisfies the identity
2

3

m

f = [z, n [x1 , y1 ], n [x1 , y1 ]p , n [x1 , y1 ]p , n [x1 , y1 ]p , . . . , n [x1 , y1]p ] ≡ 1
and therefore, by Lemma 3.5, L satisfies some non-trivial polynomial
identity. Now Zelmanov’s Theorem 3.1 implies that L is nilpotent. Let
Ĥ denote the pro-p completion of H. Then Lp (Ĥ) = L is nilpotent
and Ĥ is a p-adic analytic group by Theorem 3.2. Clearly H cannot
have a free subgroup of rank 2 and so, by Tits’ Alternative [Tit72], H
is virtually soluble. As H is residually-p we have H is soluble.
Choose arbitrarily finitely many commutators [a1 , b1 ], . . . , [at , bt ] and
m
m
let M = h[a1 , b1 ], . . . , [at , bt ]i. Since MK p is residually finite and K p
m
m
is locally graded, we conclude that (MK p )/K p is also locally graded
m
m
(Lemma 2.7). Now, looking at the quotient (MK p )/K p , the claim
m
is immediate from Lemma 2.6. As M/(M ∩ K p ) is finite we have
m
m
M ∩ K p finitely generated. In particular, M ∩ K p is nilpotent and
M is virtually nilpotent. This completes the proof.


NON-ABELIAN TENSOR SQUARE

9

Remark 4.5. The above proposition is no longer valid if the assumption of residual finiteness of G is dropped. In [DK99] G. S. Deryabina
and P. A. Kozhevnikov showed that there exists an integer N > 1 such
that for every odd number n > N there is a group G with commutator
subgroup G′ not locally finite and satisfying the identity
f = [x, y]n ≡ 1.
In particular, G′ cannot be locally virtually nilpotent. For more details
concerning groups in which certain powers are left Engel elements see
[Bas15, BS15, BSTT13, STT16].
The proof of Theorem B is now easy.
Proof of Theorem B. Let M be a finitely generated subgroup of [G, Gϕ ].
Clearly, there exist finitely many elements a1 , . . . , as , b1 , . . . , bs ∈ G
such that
M 6 h[ai , bϕi ] | i = 1, . . . , si = H.
It suffices to prove that H is virtually nilpotent. Since for every a, b ∈
G there exists a p-power q = q(a, b) dividing pm such that [a, bϕ ]q is
left n-Engel, we deduce that the power [a, b]q is left n-Engel (Lemma
2.4). That G′ is locally virtually nilpotent follows from Proposition 4.4.
Therefore, since µ(G) is the kernel of the derived map ρ′ , it follows that
H/(H ∩ µ(G)) is virtually nilpotent. As µ(G) is a central subgroup of
ν(G) we have H virtually nilpotent, which proves the theorem.

We conclude this paper by formulating an open problem related to
the results described here.
Problem 1. Let m, n be positive integers. Suppose that G is a residually finite group in which for every x, y ∈ G there exists a positive
integer q = q(x, y) dividing m such that [x, y ϕ ]q is left n-Engel in ν(G).
Is it true that G ⊗ G is locally virtually nilpotent?
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Departamento de Matematática, Universidade de Brası́lia, BrasiliaDF, 70910-900 Brazil
E-mail address: bastos@mat.unb.br
Departamento de Matematática, Universidade de Brası́lia, BrasiliaDF, 70910-900 Brazil
E-mail address: norai@unb.br

