On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field
The group UJ₂(Fq) of unitriangular automorphisms of the polynomial ring in two variables over a finite field Fq, q = pm, is studied. We proved that UJ₂(Fq) is isomorphic to a standard wreath product of elementary Abelian p-groups. Using wreath product representation we proved that the nilpotency cla...
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Інститут прикладної математики і механіки НАН України
2014
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| Цитувати: | On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field / Yu. Leshchenko, V. Sushchansky // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 2. — С. 288–297. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1529472019-06-14T01:25:01Z On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field Leshchenko, Yu. Sushchansky, V. The group UJ₂(Fq) of unitriangular automorphisms of the polynomial ring in two variables over a finite field Fq, q = pm, is studied. We proved that UJ₂(Fq) is isomorphic to a standard wreath product of elementary Abelian p-groups. Using wreath product representation we proved that the nilpotency class of UJ₂(Fq) is c = m(p − 1) + 1 and the (k + 1)th term of the lower central series of this group coincides with the (c − k)th term of its upper central series. Also we showed that UJn(Fq) is not nilpotent if n ≥ 3. 2014 Article On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field / Yu. Leshchenko, V. Sushchansky // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 2. — С. 288–297. — Бібліогр.: 7 назв. — англ. 1726-3255 2010 MSC:20D15, 20E22, 20E36, 20F14. http://dspace.nbuv.gov.ua/handle/123456789/152947 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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The group UJ₂(Fq) of unitriangular automorphisms of the polynomial ring in two variables over a finite field Fq, q = pm, is studied. We proved that UJ₂(Fq) is isomorphic to a standard wreath product of elementary Abelian p-groups. Using wreath product representation we proved that the nilpotency class of UJ₂(Fq) is c = m(p − 1) + 1 and the (k + 1)th term of the lower central series of this group coincides with the (c − k)th term of its upper central series. Also we showed that UJn(Fq) is not nilpotent if n ≥ 3. |
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Article |
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Leshchenko, Yu. Sushchansky, V. |
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Leshchenko, Yu. Sushchansky, V. On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field Algebra and Discrete Mathematics |
| author_facet |
Leshchenko, Yu. Sushchansky, V. |
| author_sort |
Leshchenko, Yu. |
| title |
On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
| title_short |
On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
| title_full |
On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
| title_fullStr |
On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
| title_full_unstemmed |
On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
| title_sort |
on the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field |
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Інститут прикладної математики і механіки НАН України |
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2014 |
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http://dspace.nbuv.gov.ua/handle/123456789/152947 |
| citation_txt |
On the group of unitriangular automorphisms of the polynomial ring in two variables over a finite field / Yu. Leshchenko, V. Sushchansky // Algebra and Discrete Mathematics. — 2014. — Vol. 17, № 2. — С. 288–297. — Бібліогр.: 7 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT leshchenkoyu onthegroupofunitriangularautomorphismsofthepolynomialringintwovariablesoverafinitefield AT sushchanskyv onthegroupofunitriangularautomorphismsofthepolynomialringintwovariablesoverafinitefield |
| first_indexed |
2025-07-14T04:24:00Z |
| last_indexed |
2025-07-14T04:24:00Z |
| _version_ |
1837594880036044800 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 17 (2014). Number 2, pp. 288 – 297
c© Journal “Algebra and Discrete Mathematics”
On the group of unitriangular automorphisms
of the polynomial ring in two variables
over a finite field
Yuriy Yu. Leshchenko and Vitaly I. Sushchansky
Abstract. The group UJ2(Fq) of unitriangular automor-
phisms of the polynomial ring in two variables over a finite field
Fq, q = pm, is studied. We proved that UJ2(Fq) is isomorphic to
a standard wreath product of elementary Abelian p-groups. Using
wreath product representation we proved that the nilpotency class
of UJ2(Fq) is c = m(p − 1) + 1 and the (k + 1)th term of the lower
central series of this group coincides with the (c − k)th term of its
upper central series. Also we showed that UJn(Fq) is not nilpotent
if n ≥ 3.
1. Introduction
Denote by UJn(F) the group of unitriangular automorphisms of the
polynomial algebra in n variables over a field F. This group over a field
of characteristic zero was studied in [1] by V. Bardakov, M. Neshchadim
and Yu. Sosnovsky. The case of n = 2 and a field of prime characteristic
was considered by Zh. Dovhei and V. Sushchansky in [3, 4].
Given a finite field Fq with q = pm elements the group UJ2(Fq) is
proved to be nilpotent and the nilpotency class has the upper bound
(q − 1)(p − 1) + 1 and the lower bound m(p − 1) + 1 [3]. Some special
subgroups of UJ2(F), where F is an arbitrary field of positive characteristic,
were described in [4].
2010 MSC: 20D15, 20E22, 20E36, 20F14.
Key words and phrases: polynomial ring, unitriangular automorphism, finite
field, wreath product, nilpotent group, central series.
Yu. Leshchenko, V. Sushchansky 289
In present paper we consider a wreath product representation of
UJ2(Fq) (section 3). Let the field Fq and its additive group be denoted
by the same symbol (it is an elementary Abelian p-group of rank m). By
Fω
q we denote the countable restricted direct power of Fq. Considering
the standard wreath product W = Fq ≀ Fω
q , where Fq is the active group,
we prove that UJ2(Fq) ∼= W (Lemma 2). Using results of H. Liebeck [5]
about nilpotent standard wreath products we obtain the nilpotency class
of UJ2(Fq).
Theorem 1. UJ2(Fq) is nilpotent of class m(p − 1) + 1.
In Section 4 we describe the central series of UJ2(Fq). We prove
Theorem 2. Let c = m(p − 1) + 1. Then the (i + 1)th term of the lower
central series and the (c − i)th term of the upper central series of UJ2(Fq)
are coincide.
Particularly, the center Z(UJ2(Fq)) is the group of all pairs
[0,
∑
i∈N
ci(x
q − x)i],
where ci ∈ Fq and ci = 0 for all but finitely many i ∈ N.
In Section 5 we consider the group UJ2(R) over an integral domain
R of a prime characteristic. It is shown that if R is a polynomial ring in
one variable over a finite field then UJ2(R) is not nilpotent (Lemma 8)
and we obtain
Theorem 3. For all n ≥ 3 the group UJn(Fq) is not nilpotent.
2. Basic definitions and notations
Let Fq be a finite field with q = pm elements. Denote by Fq[x] and
Fq[x, y] the algebras of polynomials over Fq in one and two variables
respectively. Every automorphism of Fq[x, y] is uniquely determined
by images of x and y, i.e. by a pair of polynomials 〈a(x, y), b(x, y)〉,
a(x, y), b(x, y) ∈ Fq[x, y]. An automorphism corresponding to a pair
〈αx + a, βy + f(x)〉, where α 6= 0 and β 6= 0, is called triangular. Addi-
tionally, if α = β = 1 the automorphism is called unitriangular. Then
the group UJ2(Fq) of unitriangular automorphisms of Fq[x, y] is isomor-
phic to the group of all pairs u = [a, f(x)], a ∈ Fq, f(x) ∈ Fq[x], with
multiplication
[a, f(x)] · [b, g(x)] = [a + b, f(x) + g(x + a)].
290 Unitriangular automorphisms of polynomial ring
Following [3,4] we define an elementary automorphism δa and a linear
operator ∆a, a ∈ Fq, on Fq[x] as follows:
δa(f(x)) = f(x + a) and ∆a(f(x)) = δa(f(x)) − f(x).
Then ∆0f(x) = 0 and ∆ac = 0 for every f(x) ∈ Fq[x], a, c ∈ Fq, moreover,
∆a∆b = ∆b∆a for all a, b ∈ Fq.
The identity of UJ2(Fq) is the pair id = [0, 0]. The inverse of u is
u−1 = [−a, −f(x − a)] and the commutator of u and v = [b, g(x)] is the
pair
[u, v] = uvu−1v−1 = [0, ∆a(g(x)) − ∆b(f(x))]. (1)
Lemma 1. Let f(x) ∈ Fq[x] and a ∈ Fq. Then
1) δa(xq − x) = xq − x and ∆a(xq − x) = 0;
2) δa[(xq − x)f(x)] = (xq − x)δa(f(x));
3) ∆a[(xq − x)f(x)] = (xq − x)∆a(f(x));
4)
⋂
a∈Fq
Ker ∆a = Fq[xq − x].
Proof. Parts 1), 2) and 3) can be obtained by direct computations. Let us
prove part 4). From 1) we have Fq[xq − x] ⊆ Ker ∆a for every a ∈ Fq. On
the other hand, any polynomial f(x) ∈
⋂
a∈Fq
Ker ∆a can be written as
f(x) =
t∑
i=0
(xq − x)ifi(x), (2)
where deg fi(x) < q for all i = 0, 1, . . . , t. Then, according to part 3) of
this lemma, ∆a(f(x)) =
∑t
i=0(xq − x)i∆a(fi).
Assume that there exists i such that deg fi(x) > 0. Denote g(x) =
∆a(fi(x)). Then g(0) = fi(a) − fi(0). Since fi(x) is not a constant,
there exists a ∈ F∗
q such that g(0) 6= 0. Thus, we obtain a contradiction
(∆a(fi(x)) should be 0 for every a ∈ Fq). Hence, fi(x) = const ∈ Fq for
all i = 0, 1, . . . , t.
3. UJ2(Fq) as a wreath product
The group UJ2(Fq) can be represented as a wreath product of two
elementary Abelian p-groups. We consider the standard wreath product
of Fω
q by Fq:
W = Fq ≀ Fω
q ,
Yu. Leshchenko, V. Sushchansky 291
where Fq is the active group. Elements of W are pairs [a, f(x)] such that
a ∈ Fq, f(x) is a function from Fq into Fω
q . Each such function can be
uniquely determined by the almost-zero sequence 〈f0(x), f1(x), . . .〉 of
polynomials fi(x) reduced modulo the ideal generated by xq − x, i ∈ N
(in other words, each polynomial has degree less or equal to q − 1 and
fi(x) ≡ 0 for all but finitely many i ∈ N). The identity of W is the pair
[0, 〈0, 0, . . .〉]. Now, if
u = [a, 〈f0(x), f1(x), . . .〉] and v = [b, 〈g0(x), g1(x), . . .〉] (3)
then
u−1 = [−a, 〈−f0(x − a), −f1(x − a), . . .〉];
uv = [a + b, 〈f0(x) + g0(x + a), f1(x) + g1(x + a), . . .〉]. (4)
Lemma 2. UJ2(Fq) ∼= W
Proof. Let u = [a, f(x)] ∈ UJ2(Fq) and f(x) ∈ Fq[x] has the decomposi-
tion (2). Consider a mapping ϕ : UJ2(Fq) 7→ W which acts as follows:
ϕ(u) = [a, 〈f0(x), f1(x), . . .〉] ∈ W.
Clearly, ϕ is a bijection. Now suppose v = [b, g(x)] ∈ UJ2(Fq). Then
uv = [a, f(x)] · [b, g(x)] = [a + b, f(x) + g(x + a)] =
= [a + b, f(x) + δa
(
∑
i∈N
gi(x)(xq − x)i
)
] =
= [a + b,
∑
i∈N
fi(x)(xq − x)i +
∑
i∈N
δa[gi(x)](xq − x)i] =
= [a + b,
∑
i∈N
[fi(x) + gi(x + a)](xq − x)i].
Thus,
ϕ(uv) = [a + b, 〈f0(x) + g0(x + a), f1(x) + g1(x + a), . . .〉] =
= [a, 〈f0(x), f1(x), . . .〉] · [b, 〈g0(x), g1(x), . . .〉] = ϕ(u)ϕ(v)
and ϕ is a homomorphism.
Remark 1. Elements of UJ2(Fq) can be considered as pairs of the type (3)
with group operation defined as (4). In subsequent sections we use this
representation.
292 Unitriangular automorphisms of polynomial ring
Additionally, for every i ∈ N let the projection πi : UJ2(Fq) 7→ Fq ≀ Fq
to be defined as
πi([a, 〈f0, f1, . . . , fn, . . .〉]) = [a, fi].
Obviously, πi is an epimorphism. By Wi we denote a subgroup of UJ2(Fq)
which consists of elements of the type [a, 〈0, . . . , 0, fi, 0, . . .〉], where fi is
in the ith position. It is clear that Wi
∼= Fq ≀ Fq.
In [2] G. Baumslag proved the following simple criteria: A ≀ B is
nilpotent if and only if both A and B are nilpotent p-groups, B has finite
exponent and A is finite. Additionally, H. Liebeck in [5] showed that if
B is an Abelian p-group of exponent pk, and A is the direct product of
cyclic groups of orders pβ1 , . . . , pβn , where β1 ≥ β2 ≥ . . . ≥ βn then A ≀ B
has the nilpotency class
n∑
i=1
(pβi − 1) + 1 + (k − 1)(p − 1)pβ1−1. (5)
Now, using the wreath product representation of UJ2(Fq) we can prove
Theorem 1.
Proof of Theorem 1. According to Lemma 2, the group UJ2(Fq) is iso-
morphic to the wreath product of Fω
q by Fq. Thus in terms of Formula (5)
we obtain:
1) the exponent of Fω
q equals p (i.e. k = 1);
2) the group Fq as an elementary Abelian group is the direct product
of m cyclic groups of order p (i.e. β1 = β2 = . . . = βm = 1).
Hence, c(UJ2(Fq)) =
∑m
i=1 (p − 1) + 1 = m(p − 1) + 1.
4. Central series of UJ2(Fq)
Denote by Sym(N) the group of all permutations on N = {0, 1, 2, . . .}.
Given σ ∈ Sym(N) the mapping Φσ : UJ2(Fq) 7→ UJ2(Fq) is defined as
follows:
Φσ([a, 〈f0, f1, . . . , fn, . . .〉]) = [a, 〈fσ(0), fσ(1), . . . , fσ(n), . . .〉];
in other words, Φσ permutes factors in Fω
q . Simple calculations show that
Φσ is an automorphism of UJ2(Fq) for every σ ∈ Sym(N).
Lemma 3. If K is a characteristic subgroup of UJ2(Fq) then π0(K) =
πi(K) for every i ∈ N.
Yu. Leshchenko, V. Sushchansky 293
Proof. Let us fix i. Suppose u = [a, f0] is an elements of π0(K) and
v = [a, 〈f0, f1, . . . , fi, . . .〉] ∈ K, where f1, f2, . . . , fi, . . . are polynomials
from Fq[x]/(xq − x). Consider the transposition (1, i) ∈ Sym(N). Since
K is characteristic, w = Φ(1,i)(v) = [a, 〈fi, f1, . . . , fi−1, f0, fi+1, . . .〉] ∈ K.
Hence, πi(w) = [a, f0] = u ∈ πi(K) and π0(K) ⊆ πi(K). Analogously, one
can show that πi(K) ⊆ π0(K).
The following lemma describes some properties of fully invariant
subgroups of UJ2(Fq).
Lemma 4. If a fully invariant subgroup K of UJ2(Fq) contains an element
u = [c, 〈. . .〉] with c 6= 0 then K = UJ2(Fq).
Proof. For every i ∈ N and h(x) ∈ Fq[x]/(xq − x) we define the mapping
Ψ
h(x)
i : UJ2(Fq) 7→ UJ2(Fq) as follows:
Ψ
h(x)
i ([a, 〈f0, f1, . . . , fi, . . .〉]) = [0, 〈0, . . . , 0
︸ ︷︷ ︸
i−1
, ah(x), 0, . . .〉].
Direct calculations show that Ψ
h(x)
i is an endomorphism.
Let us fix an index i and a polynomial f(x) ∈ Fq[x]/(xq − x). Since
K is fully invariant and c 6= 0, we obtain that K contains
u
f(x)
i = Ψ
c−1f(x)
i (u) = [0, 〈0, . . . , 0
︸ ︷︷ ︸
i−1
, f(x), 0, . . .〉]. (6)
Now, suppose f0(x) = g0(x)+dxq−1, where Fq ∋ d 6= 0 and degg0(x)<
q − 1. We define the endomorphism Θ : UJ2(Fq) 7→ UJ2(Fq) as follows:
Θ([a, 〈f0, f1, . . . , fi, . . .〉]) = [d, 〈0, 0, . . .〉].
Then for any given d ∈ Fq the subgroup K contains
vd = Θ([0, 〈dxq−1, 0, 0, . . .〉]) = [d, 〈0, 0, . . .〉]. (7)
Finally, elements of types (6) and (7) generate UJ2(Fq).
Corollary 1. If a fully invariant subgroup K of Fq ≀ Fq contains an
element u = [a, f(x)] with a 6= 0 then K = Fq ≀ Fq.
Lemma 5. Let V be a proper verbal subgroup of UJ2(Fq) generated by a
collection of words V and Vi be verbal subgroups of Wi with respect to the
same collection V. Then V =
∏
i∈N Vi.
294 Unitriangular automorphisms of polynomial ring
Proof. Given a word w(x1, x2, . . . , xn) ∈ V and u1, u2, . . . , un ∈ UJ2(Fq)
consider u = w(u1, u2, . . . , un) ∈ V . Then, according to the group opera-
tion (4), we obtain πi(u) = w(πi(u1), πi(u2), . . . , πi(un)) for every i ∈ N,
i.e. u is contained in the direct product
∏
i∈N Vi.
On the other hand, given v = [0, 〈f0, f1, . . . , fi, . . .〉]∈V (by Lemma 4,
since V is a proper fully invariant subgroup of UJ2(Fq), the first component
of v equals 0) we consider the element vi = [0, 〈0, . . . , 0, fi, 0, . . .〉] ∈ Vi. As-
sume vi = id for all i ≥ n. Then v = v0v1 . . . vn and vi = wi,1wi,2 . . . wi,mi
,
where wi,j , j = 1, 2, . . . , mi, is a value of a word (from V) in group Wi.
The latter implies
∏
i∈N Vi ⊆ V .
By γi(G) and ζj(G) we denote the ith and jth terms of the lower central
series and upper central series of G respectively (note that i = 1, 2, . . .,
while j = 0, 1, . . .). Given a word w = w(x1, x2, . . . , xk) and g ∈ G
let wg
i = w(. . . , xi−1, xig, xi+1, . . .) and gwi = w(. . . , xi−1, gxi, xi+1, . . .).
Recall that the marginal subgroup of G for the word w is the set of all g ∈ G
such that w = wg
i = gwi for all x1, x2 . . . , xk ∈ G and all i ∈ {1, 2, . . . , k}.
Particularly, terms of the upper central series are marginal subgroups
corresponding to simple commutators.
Proof of Theorem 2. Consider the (c − i)th member of the upper central
series as a marginal subgroup of UJ2(Fq) corresponding to the word
[. . . [x1, x2], x3], . . . , xc−i+1]. Suppose u ∈ ζc−i(UJ2(Fq)). Then, according
to (4), we obtain πj(u) ∈ ζc−i(Wj) for all j ∈ N. Thus,
ζc−i(UJ2(Fq)) ≤
∏
j∈N
ζc−i(Wj) =
∏
j∈N
γi+1(Wj).
The last equality follows from the fact that the lower central series and
upper central series of Fq ≀ Fq are coincide (for details see [7]). Since mem-
bers of the lower central series are verbal subgroups, by Lemma 5 we have
∏
j∈N γi+1(Wj) = γi+1(UJ2(Fq)). Thus, ζc−i(UJ2(Fq)) ≤ γi+1(UJ2(Fq)).
On the other hand, γi+1(UJ2(Fq)) ≤ ζc−i(UJ2(Fq)), i ∈ {0, 1, . . . , c}
(see, for example, [6], Theorem 5.31) and we obtain the result.
In particular, the center Z(UJ2(Fq)) of UJ2(Fq) is the subgroup of all
pairs
[0,
∑
i∈N
ci(x
q − x)i],
where ci ∈ Fq and ci = 0 for all but finitely many i ∈ N. In terms of the
wreath product representation Z(UJ2(Fq)) is the group of pairs
[0, 〈c0, c1, . . . , ci, . . .〉].
Yu. Leshchenko, V. Sushchansky 295
Hence, Z(UJ2(Fq)) ∼=
∏
i∈N Fq.
5. UJ2(R), where R is an integral domain
Let R be an integral domain (a non-trivial commutative ring with no
non-zero zero divisors). Denote by char(R) the characteristic of R. In this
section we assume that 1 ∈ R and char(R) = p (p is prime). Elements of
UJ2(R) are represented by pairs [a, b(x)], where a ∈ R and b(x) ∈ R[x].
Lemma 6. If u = [0, f(x)] and v = [b, g(x)] are elements of UJ2(R) then
[u, v] = uvu−1v−1 = [0, −∆b(f(x))].
The latter is a special case of Formula (1) that also holds for arbitrary
integral domain.
Assume R = Fq[ξ] is the polynomial ring in variable ξ over Fq. Now,
elements of R[x] can be considered as polynomials in variables ξ, x over
Fq. If ξkxl is a monomial then k is called the ξ-degree of the monomial
and l is called the x-degree of that monomial. Also denote
sn
m = pm + pm+1 + . . . + pn,
where m, n ∈ N (m ≤ n). We need the following technical lemma.
Lemma 7. Suppose a polynomial f(ξ, x) ∈ Fq[ξ, x] contains the monomial
ξdxsn
m such that no other term of f(ξ, x) has ξ-degree d. Then there
exists r ∈ N such that the polynomial f(ξ, x + ξr) − f(ξ, x) contains the
monomial ξd+rpm
xsn
m+1 and no other term of f(ξ, x + ξr) − f(ξ, x) has
ξ-degree d + rpm.
Proof. Let r denotes some fixed positive integer and ξdxsn
m , c1ξa1xb1 ,
c2ξa2xb2 , . . ., ctξ
atxbt are all terms of f(ξ, x); here ai, bi ∈ N, ci ∈ Fq.
Then
ξd(x + ξr)sn
m − ξdxsn
m = ξd(x + ξr)pmsn−m
0 − ξdxsn
m =
= ξd(xpm
+ ξrpm
)sn−m
0 − ξdxsn
m =
= ξdxpmsn−m
1 ξrpm
+ h(ξ, x) =
= ξd+rpm
xsn
m+1 + h(ξ, x),
where h(ξ, x) does not contain monomials of ξ-degree d + rpm.
Let us also fix i ∈ {1, 2, . . . , t}. If bi = pmidi, where p ∤ di, then by
direct computations (as in the previous case) one can show that the
polynomial ciξ
ai(x + ξr)bi − ciξ
aixbi contains only terms of the form
ξai+jrpmi xpmi (di−j), j = 1, 2, . . . , di;
296 Unitriangular automorphisms of polynomial ring
here we omit coefficients of respective monomials. Now, suppose that some
of those monomials has ξ-degree d + rpm. In other words, ai + jrpmi =
d + rpm or
r(pm − jpmi) = ai − d. (8)
If pm − jpmi = 0 then Equality (8) is false for all r ∈ N, since ai − d 6= 0.
Otherwise, if for some i and j we have pm − jpmi 6= 0 then (8) can be
rewritten as
r =
ai − d
pm − jpmi
(9)
Since t and all ai’s, bi’s are finite we can choose r such that (9) does
not hold for all possible i and j and, hence, ξd+rpm
xsn
m+1 is the unique
monomial of ξ-degree d + rpm in f(ξ, x + ξr) − f(ξ, x).
Lemma 8. If R = Fq[ξ] then UJ2(R) is not nilpotent.
Proof. Let us fix n ∈ N and u = [0, xsn
1 ] ∈ UJ2(R). We’ll prove that there
exist elements v1, . . . , vn ∈ UJ2(R) such that [u, v1, . . . , vn] 6= id.
Assume v1 = [ξr1 , 0] for some r1 ∈ N. According to Lemma 6
we obtain [u, v1] = [0, f1(ξ, x)], where f1(ξ, x) = −∆ξr1 (xsn
1 ). Here
xsn
1 satisfies the conditions of Lemma 7, thus there exists r1 such that
f1(ξ, x) contains the monomial ξr1pxsn
2 (without considering the coef-
ficient) which has the unique ξ-degree among all terms of f1(ξ, x). In
general, there exist r1, r2, . . . , ri ∈ N such that after i steps we obtain
[u, v1, . . . , vi] = [0, fi(ξ, x)], where fi(ξ, x) has a monomial of x-degree
sn
i+1 satisfying the conditions of Lemma 7. Finally, after n steps we obtain
[u, v1, . . . , vn] 6= id and the lemma is proved.
Regarding the latter lemma, it might be interesting to investigate the
necessary and sufficient conditions for UJ2(R) to be nilpotent.
Finally, we consider UJn(Fq) for n ≥ 3.
Proof of Theorem 3. Elements of the group UJn(Fq) are represented by
tuples
[a1, a2(x1), . . . , an(x1, . . . , xn−1)],
where a1 ∈ Fq and ai(x1, . . . , xi−1) ∈ Fq[x1, . . . , xi−1], i ∈ {2, 3, . . . , n}.
Let H be the subgroup of UJn(Fq) that consists of all tuples
[0, a1(x1), a2(x1, x2), 0, . . .],
where a1(x1) ∈ Fq[x1] and a2(x1, x2) ∈ Fq[x1, x2]. It is obvious that
H ∼= UJ2(Fq[x1]). Using Lemma 8 we obtain the result.
Yu. Leshchenko, V. Sushchansky 297
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Contact information
Yu. Leshchenko Institute of Physics, Mathematics and Computer
Science, Bohdan Khmelnytsky National Univer-
sity of Cherkasy, Shevchenko blvd. 79, Cherkasy,
Ukraine, 18031
E-Mail: ylesch@ua.fm
V. Sushchansky Institute of Mathematics, Silesian University of
Technology, ul. Kaszubska 23, Gliwice, Poland,
44-100
E-Mail: vitaliy.sushchanskyy@polsl.pl
Received by the editors: 22.04.2014
and in final form 22.04.2014.
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