On verbal subgroups of finitely generated nilpotent groups
The paper concerns the problem of characterization of verbal subgroups in finitely generated nilpotent groups. We introduce the notion of verbal poverty and show that every verbally poor finitely generated nilpotent group is a finite p-group with the lower p-central series for certain prime p. We c...
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | On verbal subgroups of finitely generated nilpotent groups / A. Bier // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 1–10. — Бібліогр.: 5 назв. — англ. |
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irk-123456789-1545082019-06-17T01:26:34Z On verbal subgroups of finitely generated nilpotent groups Bier, A. The paper concerns the problem of characterization of verbal subgroups in finitely generated nilpotent groups. We introduce the notion of verbal poverty and show that every verbally poor finitely generated nilpotent group is a finite p-group with the lower p-central series for certain prime p. We conclude with few examples of verbally poor groups. 2009 Article On verbal subgroups of finitely generated nilpotent groups / A. Bier // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 1–10. — Бібліогр.: 5 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20D15, 20F18.. http://dspace.nbuv.gov.ua/handle/123456789/154508 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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The paper concerns the problem of characterization of verbal subgroups in finitely generated nilpotent groups. We introduce the notion of verbal poverty and show that every verbally poor finitely generated nilpotent group is a finite p-group with the lower p-central series for certain prime p. We conclude with few examples of verbally poor groups. |
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Bier, A. |
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Bier, A. On verbal subgroups of finitely generated nilpotent groups Algebra and Discrete Mathematics |
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Bier, A. |
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Bier, A. |
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On verbal subgroups of finitely generated nilpotent groups |
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On verbal subgroups of finitely generated nilpotent groups |
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On verbal subgroups of finitely generated nilpotent groups |
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On verbal subgroups of finitely generated nilpotent groups |
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On verbal subgroups of finitely generated nilpotent groups |
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on verbal subgroups of finitely generated nilpotent groups |
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Інститут прикладної математики і механіки НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/154508 |
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On verbal subgroups of finitely generated nilpotent groups / A. Bier // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 2. — С. 1–10. — Бібліогр.: 5 назв. — англ. |
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Algebra and Discrete Mathematics |
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AT biera onverbalsubgroupsoffinitelygeneratednilpotentgroups |
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2025-07-14T06:11:49Z |
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2025-07-14T06:11:49Z |
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1837601663416795136 |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Number 2. (2009). pp. 1 – 10
c⃝ Journal “Algebra and Discrete Mathematics”
On verbal subgroups of finitely generated
nilpotent groups
Agnieszka Bier
Communicated by V. I. Sushchansky
Abstract. The paper concerns the problem of characteriza-
tion of verbal subgroups in finitely generated nilpotent groups. We
introduce the notion of verbal poverty and show that every verbally
poor finitely generated nilpotent group is a finite p-group with the
lower p-central series for certain prime p. We conclude with few
examples of verbally poor groups.
1. Introduction and main results
The characterization of verbal subgroups in a group is an interesting and
difficult problem. The full description of the verbal structure has been
found only for few specific kinds of groups. The examples are given in the
last section of this paper. All the groups presented there admit rather
poor verbal structure. We consider then an inverse problem and provide
conditions which a finitely generated nilpotent group should meet to have
such poor verbal structure.
Let G be a group and F be a set of words from the free group of count-
ably infinite rank, freely generated by an alphabet X = {x1, x2, ..., xn}.
The verbal subgroup VF (G) of group G is the subgroup generated by all
values of the words from F in group G. If F = {f} then the verbal
subgroup generated by F will be denoted by Vf (G). In a nilpotent group
every verbal subgroup is generated by a finite set of words, hence we
restrict here our considerations to the case of F being finite [3]. Let us
2000 Mathematics Subject Classification: 20D15, 20F18.
Key words and phrases: finitely generated nilpotent groups, verbal subgroups.
2 Verbal subgroups of finitely generated nilpotent groups
denote by ci the following words:
c1 = x1, ci+1 = [xi+1, ci(x1, ..., xi)].
For any group G the verbal subgroups Vci(G) constitute the lower central
series
G =
1(G) ≥
2(G) ≥ ...,
in which
i(G) = Vci(G). In the case of a nilpotent group of class l we
obviously have
l(G) = Vcl(G) = {1}.
The group is said to be verbally simple if it has no proper verbal
subgroups. In the class of residually nilpotent groups we introduce yet
another notion concerning verbal subgroups in the group. We say that
the group G is verbally poor if it has no verbal subgroups but the terms of
its lower central series. In other words group G is verbally poor if every
verbal subgroup VF (G) coincides with
i(G) for certain i ∈ N. One can
easily check that a cyclic group Cpn of order pn, where p is a prime is
verbally poor if and only if it is verbally simple, i.e. for n = 1.
In this paper we deal with the problem of finding necessary and suf-
ficient conditions for a finitely generated nilpotent group to be verbally
poor. The main result is stated in the following
Main Theorem. Every verbally poor finitely generated nilpotent group
is a finite p-group with the lower central series being a p-central series for
certain prime p.
This, however, is not a full characterization of all finitely generated
nilpotent verbally poor groups. In Section 3 we discuss the problem of
reversing the theorem and present some known examples of groups to
which the reverse theorem applies. At the moment, no counterexample
is known to the author. Then we give also a few examples of infinitely
generated verbally poor groups.
2. Preliminaries
We begin with some basic facts on verbally poor groups.
Proposition 1. Let G be a group and H be a normal subgroup of G. If
G is verbally poor then so is the quotient group G/H.
Proof. Let ' : G −→ G/H be the natural homomorphism. Since G is
verbally poor, then there exists i ∈ N such that
VF (G/H) = '(VF (G)) = '(Vci(G)) = Vci(G/H).
Hence G/H is verbally poor.
A. Bier 3
Proposition 2. Let {G1, G2, ..., Gn} be a finite family of groups and G
be the direct product of Gi, i.e. G =
n∏
i=1
Gi. If G is verbally poor then so
is Gi for every i ∈ {1, 2, ..., n}.
Proof. Consider G = A× B. Then obviously B ∼= G/A and A ∼= G/B.
Since G is verbally poor, then following Proposition 1, both A and B are
verbally poor too. The rest of the proof is simple induction on n.
From now on, we assume that G is a finitely generated nilpotent
group. We recall that
Lemma 1. [2] If G is nilpotent group and A is its subgroup such that
A[G,G] = G, then A = G.
A useful characterization of torsion-free finitely generated nilpotent
groups is known (see [2]). Namely, a finitely generated torsion-free nilpo-
tent group G has the lower central series with infinite cyclic quotients.
This fact allows us to introduce in G integer coordinates. If G = G1 >
G2 > ... > Gs+1 = {1} is the lower central series of G, then we can choose
the Malcev basis a1, a2, ..., as of group G such that Gi = ⟨ai, Gi+1⟩. Ev-
ery element x ∈ G can be uniquely represented as x = a
t1(x)
1 a
t2(x)
2 ...a
ts(x)
s
where ti(x) ∈ Z are the Malcev coordinates. The notion of Malcev basis
and coordinates in group G allows to represent elements of G as unitrian-
gular matrices with integer entries. The latter is stated in the following
Lemma 2. Every torsion-free finitely generated nilpotent group is iso-
morphic to a subgroup of UTn(Z) for a certain n ∈ N.
The group UTn(Z) is the group of upper triangular reversible ma-
trices with ones on the main diagonal, zeros below and integer entries
above. A detailed proof of Lemma 2 can be found in [2].
For the purpose of this work we also recall the notion of p-central
series. In a nilpotent group G the central series
G = �1(G) ≥ �2(G) ≥ ... ≥ �k(G) = {1}
is called a p-central series, if all the quotients �i(G)/�i+1(G) for all i =
1, 2, ..., k − 1 are elementary abelian of exponent p. An example of such
series is the lower p-central series G = �1(G) ≥ �2(G) ≥ ... ≥ �k(G) =
{1}, which is defined recursively as:
�1(G) = G, �i(G) = [�i−1(G), G]�i−1(G)p for i > 1.
Here, Gp denotes the verbal subgroup Vxp(G).
4 Verbal subgroups of finitely generated nilpotent groups
3. Proof of the Main Theorem
From the characterization of torsion-free finitely generated nilpotent groups
presented in Section 2 it follows that for description of verbal subgroups
of any torsion-free finitely generated nilpotent group, it is sufficient to
investigate the latter in the subgroups of the group UTn(Z), n ∈ N.
For further considerations we introduce some necessary notation first.
Let A = [aij ] be an arbitrary matrix from UTn(Z) such that for all indices
i < j < i+ ti + 1 there is
aij = 0 and i+ ti + 2 > n
or
ai,i+ti+2 ∕= 0.
Let w(A) = (t1, t2, ..., tn−1) be a vector of size n − 1, in which each
coordinate ti is the number of zeroes placed between the main diagonal
of the matrix A and the first nonzero element in i-th row of the matrix.
The vector w(A) will be called the type of matrix A. For example, the
type of the unit matrix In is equal to w(In) = (n − 1, n − 2, ..., 2, 1).
Directly from the given definition one can observe that
Lemma 3. If H ≤ UT(n,Z) contains matrices A and B of the types
w(A) = (t1, t2, ..., tn−1) and w(B) = (s1, s2, ..., sn−1)
respectively, then H contains a matrix C of the type
w(C) = min{w(A), w(B)}
def
= (min{t1, s1},min{t2, s2}, ...,min{tn−1, sn−1}).
Proof. It is enough to take C = AiB for adequate i ∈ N. Obviously, for
every i ∈ Z we have C ∈ H. Now, if for every k ∈ {1, 2, ..., n − 1} there
is tk ∕= sk or tk = sk ∧ Ak,k+tk+2 ∕= (Bk,k+tk+2)
−1, then for i = 1 the
assumed matrix C will be of the desired type. Otherwise, we take
i = 1 + max
k∈{1,2,...,n−1}
∣Bk,k+tk+2∣,
and this completes the proof.
As a consequence of Lemma 3 we are able to define the type WH of
the subgroup H in the group UTn(Z) as
WH = min
A∈H
w(A).
In the set of the types WH of subgroups H of the group UTn(Z) we
define the order:
(a1, ..., an) = WH1
≤ WH2
= (b1, ..., bn) ⇔ ∀i = 1, ..., n ai ≤ bi.
A. Bier 5
If WH1
≤ WH2
and there exists i ∈ {1, 2, ..., n} such that ai ∕= bi, then
we write
WH1
< WH2
.
Lemma 4. Let
i(H), i = 2, 3, ..., c be the i-th term of the lower central
series in the group H being the subgroup of UTn(Z) of nilpotency class c.
Then
W
c(H) > W
c−1(H) > ... > W
2(H) > WH .
Proof. If W
i(H) = (t1, t2, ..., tn−1), then W
i+1(H) = (t1+1, t2+1, ..., tn−1+
1).
Lemma 5. Let w(A) = (t1, t2, ..., tn−1) be the type of the matrix A. Then
w(Ak) = w(A) for every k ∈ Z ∖ {0}.
Proof. Indeed, it can be easily observed that
(Ak)i,i+ti+2 = k ⋅Ai,i+ti+2 ∕= 0
for k ∕= 0. Moreover, if i < j < i + ti + 1 then (Ak)ij = 0, and hence
w(Ak) = w(A).
Lemma 6. The group UT(n,Z) does not contain divisible subgroups
different from {In}.
Proof. Let us assume that H is a nontrivial divisible subgroup of UT(n,Z).
Then there exists A ∈ H such that A ∕= In, ie. w(A) = (t1, t2, ..., tn−1)
and there exists i0 ∈ {1, 2, ..., n − 1} such that ti0 < n − i0. Since
Ai0,i0+ti0+2 ∈ Z ∖ {0}, then there exists m ∈ Z ∖ {0} such that m ∤
Ai0,i0+ti0+2. Indeed, it is enough to take m > ∣Ai0,i0+ti0+2∣. If there
exists a matrix B such that Bm = A, then following Lemma 5 we have
w(B) = w(A) and the equality
Ai0,i0+ti0+2 = m ⋅Bi0,i0+ti0+2
holds. Then we get a contradiction and hence there exists no such matrix
B. Therefore H is not a divisible group.
Proposition 3. Every verbally poor finitely generated nilpotent group is
a (finite) torsion group.
Proof. The proof will be carried out in a few steps. Successively, we will
show the following:
A If H is a non-trivial subgroup of group UTn(Z), then for every
k ∕= 0,±1 the verbal subgroup Vxk(H) is a proper subgroup of H.
6 Verbal subgroups of finitely generated nilpotent groups
B A non trivial subgroup of UTn(Z) is not a verbally poor group.
C Correctness of the thesis in the Proposition.
A. The first part of the proof consists of two steps:
1. At first we will show that if A ∈ Vxk(H) and w(A) = (t1, t2, ..., tn−1),
then the elements Ai,i+ti+2 for i = 1, 2, ..., n− 1 are divisible by k.
Indeed. Let A be a matrix from Vxk(H). Then there exist ma-
trices A1, A2, ..., As ∈ H such that A = Ak
1A
k
2...A
k
s . If w(Ai) =
(u
(i)
1 , u
(i)
2 , ..., u
(i)
n−1) for i = 1, 2, ..., s, then
w(A) = min
i
w(Ai)
and
Aj,j+tj+2 =
s∑
i=1
k ⋅Aj,j+tj+2 = k ⋅
s∑
i=1
Aj,j+tj+2 □
2. Now, assume Mi ⊂ Z to be defined as follows: Mi
def
= {Ai,i+ti+2∣A ∈
H}. Since H ∕= {1n}, then there exists i0 ∈ {1, 2, ..., n − 1} such
that Mi0 ∕= {0}. Let us denote by m the element of the smallest
nonzero absolute value in Mi0 (if there are more than one such
elements we choose one of those that are positive numbers). The
following two cases may occur:
a) k ∤ m. Then there exists matrix A ∈ H such that Ai0,i0+ti0+2 =
m and following the first step of the proof A ∕∈ Vxk(H).
b) m = k ⋅ l for certain l ∈ Z ∖ {0} such that ∣l∣ < ∣m∣. Let us
assume that matrix A ∈ H satisfying condition: Ai0,i0+ti0+2 =
m is contained in Vxk(H). Then there exist matrices A1, A2, ..., As ∈
H such that
A = A(1)kA
(2)k
2 ...A(s)k
s , s > 1.
Hence m = k ⋅ l = k ⋅
s∑
i=1
A
(i)
i0,i0+ti0+2, and therefore
l =
s∑
i=1
A
(i)
i0,i0+ti0+2 ∈ Mi0 .
By the fact that ∣l∣ < ∣m∣ and the assumptions involving m we
get a contradiction. Hence A ∕∈ Vxk(H).
A. Bier 7
From a) and b) we directly conclude that Vxk(H) ∕= H for k ∕=
0,±1.
B. Now we can prove the second part. If A ∈ H is a matrix of the
type WH , then after Lemma 5 the matrix Ak, k ∕= 0 is a matrix of the
type WH , hence the type of the verbal subgroup Vxk(H) of the group H
generated by the word xk is equal WH . Simultaneously if k ∕= ±1, then
Vxk(H) ∕= H.
We obtained that while k ∕= 0 the inequalities hold: WV
xk
(H) = WH <
W
i(H) for i = 2, 3, ..., c. Hence
Vxk(H) ∕=
i(H) dla i = 2, 3, ..., c,
i.e. group H has verbal subgroups Vxk(H), that do not coincide with any
term of its lower central series.
C. Let G be a finitely generated nilpotent group with non-trivial torsion-
free part. We denote by T the torsion part of G. Of course, T ⊲ G
and the quotient group G/T is torsion-free. Then G/T is isomorphic
to a subgroup of UTn(Z), which - as proved above - is not a verbally
poor group. Hence by Proposition 1, neither is G. Therefore, a finitely
generated verbally poor nilpotent group G has a trivial torsion-free part,
i.e. it is a finite torsion group.
Proof of the Main Theorem. As a consequence of Proposition 3,
the research of verbally poor finitely generated nilpotent groups can be
limited to finite torsion groups. The structure of finite nilpotent groups
has been thoroughly investigated and we recall here a famous result of
Burnside and Wielandt, that a finite nilpotent group is a direct product
of its maximal p-subgroups. Then from Proposition 2 we obtain a simple
observation, that if a finite nilpotent group G has a maximal p-subgroup
which is not verbally poor, then neither is G.
Let us consider the case of G being the direct product of groups A
and B such that
exp(A) = pk, exp(B) = ql, p ∕= q,
where p and q are two different primes. We will show that G is not
verbally poor. Obviously
V
xpk (A) = {1A} and V
xql (B) = {1B}.
Also, since LCD(pk, ql) = 1 then V
xql (A) = A and V
xpk (B) = B, hence
V
xpk (G) = {1A} ×B and V
xql (G) = A× {1B}.
8 Verbal subgroups of finitely generated nilpotent groups
However, the subgroups {1A}×B and A×{1B} do not coincide with any
of the terms of the lower central series in G. As A and B are nilpotent,
then
i(A) ∕= A and
i(B) ∕= B for i > 1 and therefore
A× {1B} ∕=
i(G) ∕= {1A} ×B.
For i = 1 we have
i(G) = G and this term also does not coincide with
the verbal subgroups determined above. Hence G is not verbally poor, if
it is not a p-group.
Now, consider a verbally poor finite p-group G of nilpotency class c.
As a p-group, G has the lower central series with abelian sections being
p-subgroups. We will show that the lower central series of G is a p-central
series, that is
i(G) = �i(G) for all i = 1, 2, ..., c+ 1.
The proof is inductive. We start with �2(G) = G′Gp and note that
the second term of the lower p-central series of group G is the Frattini
subgroup Φ(G) of G. It is well known that
2(G) = G′ ⊆ Φ(G), hence
we need only to prove the reverse inclusion. It is enough to show that
Gp ⊆ G′. Since G is verbally poor, then Gp coincides with one of the
terms of the lower central series of G, say Gp =
l(G) for certain l.
Moreover, since G is a finite p-group, Gp is a proper subgroup of G and
l ≥ 2. Then Gp =
l(G) ⊆
2(G).
Now, assume that
i(G) = �i(G) for all i ≤ k. We take
�k+1(G) = [�k(G), G]�k(G)p.
By induction it is equal to [
k(G), G]
k(G)p =
k+1(G)
k(G)p. We need
only to show that
k(G)p ⊆
k+1(G). Indeed,
k(G)p is a proper verbal
subgroup of
k(G) and hence it is a verbal subgroup in G. Since G is
verbally poor,
k(G)p =
m(G) for certain m ≥ k + 1 and in particular
k(G)p ⊆
k+1(G).
This completes the proof of Main Theorem.
4. Discussion
The Main Theorem describes the structure of a finitely generated nilpo-
tent verbally poor group, however it does not provide a full character-
ization. It is an interesting question whether the reverse of the Main
Theorem holds and the p-central lower central series is a sufficient con-
dition for verbal poverty. Although the answer is not totally clear, the
reverse theorem seems to be true. This hypothesis can be easily verified
for the abelian and metabelian groups.
A. Bier 9
Remark 1. A finite abelian group G is verbally poor if and only if G is
elementary abelian.
For metabelian groups we have:
Proposition 4. If a finite metabelian p-group G has the lower central
series which is a p-central series, then G is verbally poor.
Proof. We assume that G has the lower central series
G ≥ G′ ≥ {1}
such that G/G′ ∼= Ck
p and G′/{1} ∼= G′ ∼= Cm
p . We denote by ' the
natural homomorphism of G onto G/G′. Let VF (G) be an arbitrary verbal
subgroup of G. Then '(VF (G)) = VF (G/G′) and since G/G′ ∼= Ck
p is
verbally simple, then there are two options:
1. VF (G/G′) = G/G′. Then VF (G) contains representatives of all
cosets of G/G′ and we have G = VF (G) ⋅ G′. Following Lemma 1,
VF (G) = G.
2. VF (G/G′) = 1 ⋅G′. Then VF (G) ≤ G′, and since in G′ ∼= Cm
p there
are no nontrivial fully invariant subgroups, then VF (G) = {1} or
VF (G) = G′.
Overall, the only verbal subgroups of G are G, G′ and {1}, i.e. G is
verbally poor.
So, for finite nilpotent groups of nilpotency class less than 3, the Main
Theorem defines all verbally poor groups. Apart from all such groups,
we give another examples of finite nilpotent verbally poor groups.
Example 1. The Sylow p-subgroup Sylp(Sn) of a finite symmetric group
is verbally poor. The proof and details can be found in [5].
Example 2. The group Wn defined as a wreath product Wn = Cn
p ≀Cn
p ≀
... ≀ Cn
p is verbally poor group.
This paper addresses only finitely generated nilpotent groups, however
there exist verbally poor groups with infinite number of generators. We
provide here three examples of such groups:
Example 3. The group of automorphisms AutT2 of the homogeneous
2-adic rooted tree is verbally poor [4].
Example 4. The Sylow p-subgroup of the group of automorphisms
AutTp of a p-adic rooted tree for p > 2 is a verbally poor group.
10Verbal subgroups of finitely generated nilpotent groups
Example 5. The group UTn(K) of unitriangular matrices of size n
over a field K of characteristic 0 is a verbally poor group. Every verbal
subgroup of UTn(K) coincides with one of its subgroups of the type
UT l
n(K), 0 ≤ l ≤ n− 1 (see [1] for details).
It is an interesting question, whether any conditions for verbal poverty
can be found for the infinite nilpotent (or residually nilpotent) groups.
References
[1] Bier A., "Verbal subgroups in the group of triangular matrices over a field of
characeristic 0", J. Algebra vol. 321 nr 2(2009), p. 483-494
[2] Kargapolov M. I., Merzlyakov Yu. I., "Fundamentals of the theory of groups",
Springer, 1979.
[3] Neumann H., "Varieties of groups", Springer-Verlag New York, 1967.
[4] Smetanyuk, N., Sushchansky V., "Verbal subgroups of the finitary automorphism
group of a 2-adic tree", Fundam. Prikl. Mat. 6 (2000), no. 3, p. 875-888
[5] Sushchansky V., "Verbal subgroups of the Sylow p-subgroups of the finite sym-
metric group", Bull. Kyiv University, ser. Math. and Mech. vol. 12 (1970), p.
134-141
Contact information
A. Bier Institute of Mathematics, Faculty of Math-
ematics and Physics, Silesian University of
Technology, Poland
E-Mail: Agnieszka.Bier@polsl.pl
Received by the editors: 15.05.2009
and in final form 08.10.2009.
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