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5
Journal of Siberian Federal University. Mathematics & Physics 2018, 11(1), 66—69
УДК 512.5
On a Question about Generalized Congruence Subgroups
Vladimir A. Koibaev*
North-Ossetian State University Vatutin, 44-46, Vladikavkaz, 362025 Southern Mathematical Institute VSC RAS Markus, 22, Vladikavkaz, 362027
Received 17.04.2017, received in revised form 20.05.2017, accepted 22.10.2017 Elementary net (carpet) a = (atj) is called admissible (closed) if the elementary net (carpet) group E(a) does not contain a new elementary transvections. This work is related to the problem proposed by Y.N.Nuzhin in connection with the problem 15.46 from the Kourovka notebook proposed by V.M.Levchuk (admissibility (closure) of the elementary net (carpet) a = (aij) over a field K). An example of field K and the net a = (aij) of order n over the field K are presented so that subgroup {tij (aij),tji(aji)) is not coincident with group E(a) П {tij (K), tji(K)).
Keywords: Carpets, carpet groups, nets, elementary nets, allowable elementary nets, closed elementary nets, elementary net group, transvection. DOI: 10.17516/1997-1397-2018-11-1-66-69.
1. Notations and problem statement
Let us consider problem 15.46 from the Kourovka notebook [1] on the admissibility (closure) of carpets (elementary nets) proposed by V. M. Levchuk. This problem (or rather, its SL-version) is as follows. Let a = (aij) be an elementary net (carpet) of order n > 3 over a field K. Is it true that for the admissibility of the carpet (elementary net) a = (a j), 1 < i = j < n is necessary and
sufficient the admissibility of subcarpets (subnets) ( * aji\ of second order (for any i = j)?
\aij * J
We note that solution of this problem for locally finite fields results from [2]. In connection with the theorem on the decomposition of the elementary transvection in the elementary net group E(a) [3], a sufficient condition to solve the problem was proposed by Y.N.Nuzhin. It is linked with the validity of the equality
E(a) n(tij(K), tji(K)> = (tij(aij), tji(aji)) (1)
for all i = j, where a = (aij) is the closed elementary net of degree n > 3 over a field K, E(a) is the elementary net subgroup. Inclusion (D) is obvious. To test the validity of equality (1) one need to test the validity of inclusion (C) in (1).
In this paper we present an example of field K and elementary closed (admissible) irreducible net a = (aij) of order n > 3 over the field K for which the subgroup E(a) n (tj(K), tji(K)> is not contained in the group (tij(aij), tji(aji)>. Without the loss in generality we assume that i = 1, j = 2. The proposed example results from [4, 5]. We should note that solution of problem 15.46 from the Kourovka notebook is not presented in the paper.
* [email protected] © Siberian Federal University. All rights reserved
In the paper the following standard notations are adopted: Sij is the Kronecker delta; tij (a) = e + aeij is the elementary transvection, where e is the identity matrix of order n, ej is the matrix, its entries at (i,j) are equal to 1, and all other entries are equal to zero, a e K; tij(A) = {tij(a) : a e A};
E(a) = (tij (oij) : 1 < i = j < n) is the elementary net group defined for an elementary net a = (aij);
Eij(a) = (tij(aij), tji(aji)), i = j;
F is arbitrary commutative ring with 1; F [x] is the ring of polynomials with respest to one variable x with coefficients from F; K = F(x) is the the field of all rational functions
-,f,g e F[x], g = 0. g
2. Deriviation of the example
To begin with, recall well-known definitions that we use in this paper. A set of additive subgroups a = (aij), 1 < i,j < n, of a field (or ring) K is called a net of order n over K if airarj C aij for all values of i, r,j [6]. The term carpet is also used instead of the term net [7, 8]. The same system but without the diagonal is called elementary net. A full or elementary net a = (aij) is called irreducible if all additive subgroups aij are different from zero. An elementary net a is closed (admissible) if the subgroup E(a) does not contain new transvections. If the diagonal of an elementary net can be supplemented by a subgroup to a full net then such elementary net is closed.
For a non-negative integer n e N U 0 consider the ideal
Fn[x] = {cnxn + cn+1xn+1 + ... + cmxm : m > n, ci e F} of ring F[x] = F0[x]. It is obvious that (n, s e N U 0)
Fn [x]Fs [x] = Fn+S [x], Fn [x] D Fn+1 [x] D Fn+2 [x].... (2)
Let us consider the supplemented (in particular, closed) elementary net of order n > 3)
/ * F2[x] Fi [x]
F2[x] * Fi [x]
Fi[x] Fi[x] *
\Fi [x] Fi[x] Fi [x]
Fi[x]\ Fi[x]
Fi[x] *
(3)
of ideals of the ring F[x]: a12 = a21 = F2 [x], aij = F1[x] for other i = j. Taking into account (2), the table of a is an elementary net over ring F[x] ( or over field K = F(x)) of order n. This elementary net is supplemented net (for example, F1 [x] can be put in all positions of the diagonal). For elementary net a (3) consider the subgroup
H = (tij(aij) : 1 < i = j < n; {i,j} = {1, 2})
of the elementary group E(a) generated by all root subgroups tij(aij) except of t12(a12) and t21(a21).
a =
Proposition. Elementary net group E(a) is equal to the product of the group E 12(a) and the normal subgroup H: E(a) = E 12(a) ■ H.
Proof. If a G E(a),a is the product of elementary transvections of E(a), then sequentially pulling elementary transvections t12(*) and t21(*) to the left, we get the inclusion a G E12(a) ■ H. Let us show that H is an normal subgroup of the group E(a). Taking into accoun the equality E(a) = E12(a) ■ H, it is sufficient to show that shs-1 G H for all s G E12(a), h G H. Let assume that i ^ 3. Then we have
ti2(a)t2i(@)t-2(a) = tu(aP)t2i(P), t2i(a)t1i(ft)t-li(a) = t2i (aft)tu (ft).
□
Theorem. The subgroup H n (t12(K), t21(K)) is not contained in the group E12(a). In particular, the subgroup E(a) n (t12(K), t21(K)) is not contained in the group E12(a) = (t 12(a 12), t21(a21)).
Proof. Let us assume that a, ft, 7, S are elements of arbitrary commutative ring and aS+7ft = 0, ([z,t] = ztz-1t-1). Then
b = [t23(a)t13(ft), t31(Y)t32(S)] = diag( ^ ^ft1 1 ^^ , e
(general transvection). Applying this formula to our case, we put a = -x1, ft = 7 = S = x1. Then matrix
b = [t23 (-x)t13(x), t31 (x)t32(x)]
has the form
b = diag( i^lS 1 tx*) , en-2) .
Hence matrix b is contained in the group H n (t12(K), t21(K)). □
To prove the theorem one need to show that matrix b is not contained in the group (t12(a12),t21(a21)). This follows from the following lemma.
Lemma. If c =(Sj + Cj) G (P12(au), t21(a21)) then 012,021 G F2[x] and 011,022 G F4[x].
Proof. Let matrix c be the product of n elementary transvections of t12(F2[x]) and t21(F2[x]): c = t12(a1 )t21(a2)t12(a3)... . The proof of the Lemma is carried out by induction on n, and induction transition follows from (2) and inclusions (a,a12,a21 G F2[x], a11,a22 G F4[x])
'1 + au a 12 )(1 a\ (1 + F4[x] F2[x] a21 1 + a22j V0 1) "V F2[x] 1 + F4[x],
(1 + an a 12 \ (1 a)c(1 + FM F2[x] \
V a21 1 + a22) V° l) V F2[x] 1 + F4[x]J '
(1 + an a 12 )( 1 0\c(1 + Fa[x] F2X] \
V a2i 1 + a22j Va V V F2[x] 1 + F4[x]J '
'1 + an a12 \ (1 + Fi[x] F2[x]
a21 1 + a22j Va 1) V F2[x] 1 + F4[x]^
□
Author is grateful to Ya. N. Nuzhin who proposed the problem discussed in the paper.
The work was supported by the Ministry of Education and Science of the Russian Federation and by Southern Institute of Mathematics (Vladikavkaz Scientific Centre of Russian Academy of Sciences).
References
[1] The Kourovka notebook: unsolved problems in group theory, Russ. acad. of sciences, Siberian div., Inst. of mathematics, Novosibirsk, Issue 17, 2010.
[2] V.M.Levchuk, On generating sets of root elements of Chevalley groups over a field, Algebra i logika, 22(1983), no. 5, 504-517 (in Russian).
[3] R.Y.Dryaeva, V.A.Koibaev, Decomposition of elementary transvection in elementary group, Zapiski Nauchnykh Seminarov POMI, 435(2015), 33-41 (in Russian).
[4] V.A.Koibaev, Nets associated with elementary nets, Vladikavkaz. Mat. Zh., 12(2010), no. 4, 39-43 (in Russian).
[5] V.A.Koibaev, Elementary nets in linear groups, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 17(2011), no. 4, 134-141 (in Russian).
[6] Z.I.Borevich, Subgroups of linear groups rich in transvections, Journal of Soviet Mathematics, 37(1987), no. 2, 928-934.
[7] V.M.Levchuk, Parabolic subgroups of certain ABA-groups, Mat. Zametki, 31(1982), no. 4, 509-525 (in Russian).
[8] V.A.Koibaev, Y.N. Nuzhin, Subgroups of the Chevalley Groups and Lie Rings Definable by a Collection of Additive Subgroups of the Initial Ring, Journal of Mathematical Sciences, 201(2014), no. 4, 458-464 .
К вопросу об обобщенных конгруэнц-подгруппах
Владимир А. Койбаев
Северо-Осетинский государственный университет Ватутина, 44-46, Владикавказ, 362025 Южный математический институт ВНЦ РАН Маркуса, 22, Владикавказ, 362027
Россия
Элементарная сеть (ковер) а = (а^) называется допустимой (замкнутой), если элементарная сетевая (ковровая) группа Е(а) не содержит новых элементарных трансвекций. Работа связана с вопросом, поставленным Я.Н.Нужиным в связи с вопросом В.М.Левчука 15.46 из Коуровской тетради о допустимости (замкнутости) элементарной сети (ковра) а = (а^) над полем К. Приводится пример поля К и сети а = (а^) порядка п над полем К, для которой подгруппа (аЧ )> не совпадает с груППой Е(а) П(1ц(К), Ьц(К)).
Ключевые слова: ковры, ковровые группы, сети, элементарные сети, допустимые элементарные сети, замкнутые элементарные сети, элементарная сетевая группа, трансвекция.
