GENERATORS AND RELATIONSHIPS IN GENERALIZED M-TRIANGULAR GROUPS OVER AN ASSOCIATE RING. I Текст научной статьи по специальности «Математика»

GENERATORS AND RELATIONSHIPS IN GENERALIZED M-TRIANGULAR GROUPS OVER AN ASSOCIATE RING. I

1Satarov Zhoomart, 2Mamaziaeva Elmira, 3Mambetov Zhoomart, 4Orunbaeva N.A.

1Doctor of Physical and Mathematical Sciences, Professor Osh Technological University named after the M. Adyshev,Kyrgyzstan, Osh.

2Candidate of Physical and Mathematical Sciences, Associate Professor: Osh State University,Kyrgyzstan, Osh.

3Candidate of Physical and Mathematical Sciences, Associate Professor Osh Technological University named after the M. Adyshev,Kyrgyzstan, Osh. 4Osh State Pedagogical University named after the A.J.Myrzabekova,Kyrgyzstan, Osh.

https://doi.org/10.5281/zenodo.10062871

Abstract. The question of representing linear groups (and related constructions) by generating elements and defining relations has always been of interest in general combinatorial group theory. Today, a large amount of magazine and book materials have already accumulated in this direction. New research methods also emerged. One of them is the universal combinatorial transformation method, the essence of which is to transform words of the selected generative alphabet of the group under study to their standard forms. The paper provides a description through generators and defining relations of generalized m-triangular groups T°nm (R), n > 2 defined over an arbitrary non-zero associative ring. Based on this result, combinatorial descriptions of the projective factors of the named groups PT°m (R) are also found. The solution

to these problems is based on the mentioned transformation method.

Keywords: generators, relations, quasi-multiplication, quasigroup, generalized m-triangular group, standard forms, transformation of letters, completeness of relations, projective factor.

INTRODUCTION

The representation of linear (and close to them) groups in terms of generators and relations is one of the main issues in combinatorial group theory. This section has long grown into a special direction in general theory and is currently experiencing rapid development. Within the framework of this topic, we can note the remarkable (and already classic) results [1]-[4]. The proposed work is also devoted to the named section, or rather, here we will give a combinatorial description of generalized m-triangular groups of degree n > 2 over an arbitrary associative ring.

Throughout, we assume an arbitrary nonzero associative ring for which the existence of 1 is not necessary. Through °, as always, we denote quasi-multiplication in R T.e. x o y = x + xy+y for elements x,yeR . Element x from is called quasi-invertible if for it

X 0 y = 0 = y 0 x at some y e R . Given a quasi-reversible, its quasi-inverse is always determined uniquely and it is denoted as y = x'. The set of all quasi-invertible elements R0 from R is nonempty (for example 0e R0) and it forms a group relative to the operation o . Unit in R0 element 0 will serve. We call this group the quasigroup of the ring.

In the special case, putting instead the ring of (upper) triangular matrices Tn (R), we come to the concept of a generalized triangular group [Tn (R)]° = TO (R) degrees n above the ring R . For natural m, 1 < m < n, by analogy with [5] (see p. 24) we denote by T0m (R) set of matrices from TO(R) from m-1 zero diagonals above the main one, i.e.

TOm (R) = {x = (x,) e TO (R): o < j - i < m ^ xv = 0} .

Let us show that the introduced sets form subgroups T0>m (R). To do this, we just need to check the closedness T0>m (R) with respect to matrix quasi-multiplication and the operation of taking a quasi-inverse element. Let, along with the above x = (xtJ) put from TOm (R) another matrix y = (y^). As is easy to see, for positions (i, j, 0 < j - i < m, quasi-products of these matrices satisfy the formulas (x ° y) j = xij xlky,j + yj = £ xlky,j .

1< x<n 1< x<n

Since when k ^ i 0 < j - i < m & i < k < j ^ 0 < k - i < m, first factors xA, i < k < j, the last amount will be equal to zero. When k = i we have y^ = 0. Thus, the equalities (x ° y)tj = 0 true for all the above positions (i, j), those isolation in T0m (R) occurs.

To continue our reasoning further, we need the following notation: for s e R° dt (£■) -matrix of TO (R), differing from the zero matrix only by position (i,i), where is the element s; the same way ^ (A), i ^ j, will mean matrix (also from TO (R)), obtained from the zero matrix by replacing its position (i, j for argument Ae R (they are called quasi-transvections). For the introduced matrices the formulas are obvious: di (s) = dt (s1), tj (A) = tjj (-A).

0

Just now x- arbitrary matrix of T0>m (R). From equality (sf) of this work (see paragraph I) it follows thatx' = f1 °... ° f0-m ° d'nSn) °... ° d/(Si)

(fi - some words composed of quasi-products of transvections of the form ^ (Ak)). Application to the right side of the last equality of relations () will lead us to a representation of the matrix x/ consisting of a quasi-product of (a finite number of) diagonal letters d, (s) and quasi-transvections ^ (*). And this, according to the closedness already established above, means belonging to Tnom (R) not only x, but also its quasi-inverse matrix x7. So, group inclusion Tno,m (R) < TOO (R) we have completely installed it. Entered group T0m (R) we will call the generalized m-triangular group of degree n > 2 over the ring R. As noted above, our main goal in this part of the work is to define in terms of generators and relations of triangular groups T0m (R) m = 1,2,..., n . It is carried

out exactly the same for all specified values. m. Entered groups in TO (R) form a descending chain

TO(R) = TOi(R) >TnO (R) >... >TOn(R) = DO(R)

' n \ J n,2\J "' n,n\ s n

(where D0(R)- R0 x... x R0 - diagonal in TO(R) ). A similar serial description was carried out

n раз

earlier in [6] for subgroups of the complete linear group GLn (A), n > 2, over the local ring A (with

small restrictions on A), containing a group of diagonal matrices Dn (A). In concept, our work is

also close to work [7], where the combinatorial structure of a triangular group of any (even infinite) order was studied.

It is easy to see that if there is a 1 in R, the mapping

T m(R) ^ T° (R), e + x ^ x

n,m\ ' n,m v /?

(e- unit order matrix n), defines an isomorphism of groups. Therefore, the groups introduced above (R) are generalizations of the usual m-triangular groups (respectively) to the most general

cases of associative rings R. When solving the problem, we again use the transformation method developed in [9] and [10].

Standart forms in T°m (R)

They are defined relative to some generating system of the named group. As such we take the system

dk (e), e g Ro, 1 < k < n; tj (A), A g R, j-i > m. (g)

The fact that the group T°m (R) is generated by the alphabet (g), follows directly from Theorem 1 of this paper. Under the step forms i here we understand words of the form f = ^t!/t(Ak ) (where multiplication is quasi-multiplication and the order of the factors is

likv

i+m<x<n

unimportant). As standard forms, all possible combinations of the alphabet (g) of the form are declared herex = dx{sx) °... о dn{sn) о fn m °... ° f (sf)

(where m=n expression fn_m °... ° f meaning is given 0).

Regarding the entered forms, it occurs

Theorem 1. Any matrix x from T^m (R), n > 2, presented in standard form (sf), and such a representation is unique.

Proof. Uniqueness. Just f = tlm+l(Лт+1) °... ° tin (A ). Here

d2(s2) °... ° dn(sn) ° fn_m °... ° f has a cell-diagonal appearance diag(0,xx), dx(sx) ° f has the same first row as x, i.e.. xn,0,...,0,17..., xln. Equating the corresponding positions here gives us £1 = хп,Лт+1 x1m+1 + xUm+1,..., Лп =s! x1n + x1n, т.е. s1 и f matrice x are determined unambiguously. Moving now from x to the matrix

d[ (sx) ° x ° f( = d2(s2) °... ° dn(sn) ° fn_m °... ° f, we similarly conclude the uniqueness s2 and f2. The process described on (n-m)-M step leads us to the conclusion about the uniqueness sn_m and /и_т. And then the equalities sk = хи, n - m < к < n, already take place in an obvious way.

As for the existence part of the theorem, it is a direct consequence of Theorem 3 of this paper. Therefore, we can omit it here too. The case m=n can also be included in this theorem, if we assume that there°... ° f = 0.

Constitutive relations In the alphabet (g) we can write the following (directly verifiable) group relations T°m (R):

1. di (s) ° di (a) = di (s °a);

2. dt (s) ° dk (a) = dk (a) ° di(s), i ф к;

3. t k (A) ° t ik (a) = tlk (A + a);

4. t,k(A) ° tj (a) = tj (A) ° tk (a), k * r, i * j;

5. tk (A) ° tj (a) = tj (Aa) ° tj (a) ° tA (A);

6. t,k (A) ° dt (s) = dt (s) ° tlk (A + s'A);

7. tlk(A) ° dk (s) = dk (s) ° tk (A + As);

8. tlk (A) ° dr (s) = dr (s) ° tk (A); r * i, k.

Our immediate goal is to show the completeness of the system of relations 1-8 for the

group TOm (R) in generating (g). For this purpose, we introduce (binary) relations on the set of all

i i words of the alphabet (g) 1 < i < n - m, put in W^V if and only if the words W and V

related by the relation W = X ° V, where X- some word that does not contain non-zero quasi-

i

transvections of the form t^ (*), k < i. How to easily check entered relationships ^ are reflexive

and transitive.

Next, we will need the following

Theorem 2 (about the transformation of letters). Let f- some form of step i and x- nonzero letter of the alphabet (g), for which x = t (A) condition is considered fulfilled p > i. Then

i

for them, using relations 3-8, you can perform the transformation V = f ° x ^gi, where g- also some form of stage i.

The proof is combinatorial and is carried out in two stages. Below we, to simplify the entries under f (* r) let's agree to understand the form fi, not containing a letter of the form tir (*), ** 0 .

Stage I. x = dk (s)

Here, using relations 6-8, we will have

V = f (* n) ° [tin (A) ° dk (s)] = [f (* n) ° dk (s)] ° t^ (A). Continuing this movement dt (s) and further, we arrive at the required form like this

V = dk (s) ° t,,,+m (*,+m ) ° ... ° tin (*n ) ^ ti.i+m (*i+m ) ° ... ° tin (*n ) = gt.

Stage II. x = t^ (A).

Here our consideration branches out as follows.

a)r=i. Applying relations 4 and 3, here we obtain the required form as follows

V = ft (* j) ° [tj (*) ° t j (A)] = [f (* j) ° t j (* + A)] = g i.

e) r>i. In this case, using relations 4 and 5, we will have

V = f (* r) ° [tr (*) ° tj (A)] = [f (* r) ° tj (A)] ° tj (a) ° tr (*) = tj (*) ° f (* r) ° tj (*) ° tr (*) ^. [ft (* r) ° t j (*)] ° t r (*) .

The resulting word by applying the already analyzed point a) to the selected segment leads

i

us to the required form as V ^ f (* r) ° tir (*) = gi. Theorem 2 is proven.

3. Group View TOm (R)

We are now ready to formulate a basic statement about the representation of the named

group.

Theorem 3. Generalized m- triangular group T^ (R), n > 2 (1 < m < n), over the associative ring R ^ {0} in generators (g) is represented by relations 1-8. The proof consists of two parts. I. Reduction to standard form.

In this part we will show the reducibility of any word W of the alphabet (g) to its standard form S(W) using relations 1-8. Without loss of generality, a given word can be considered represented in the form

W—f ° X, (0)

Where fi- some form of stage 1 and X is its corresponding complement. Let further, = x o X1 x- the first letter of the complement X. Applying transformation theorem 2 (i.e. using

i

relations 3-8), we reduce the given word to the form W = f o x] o X — ^ o Xx,. we get a

notation of the same form (0), but with a shortened complement X. Continuing this reduction

i

further (until all X is exhausted), we come to a notation of the form W—f (where fi- another

i

form of step 1).Last according to definition — means that W = Y o f, where is the (already left) complement 71 does not contain quasi-transvection tly. (*), * ^ 0. Now we do the same with Y1 and extract the shape from itf2(cryneHH 2), we have W = Y2 o f o f, where is the complement Y2 does not contain a quasi-transvection of the form tft (*), * ^ 0, i < 2, etc. The described process of form splitting off at the (n-m)th step leads us to the notation

W = Y o f o ... o f o f ,

n-m J n-m J 2 J 1 ?

n-m

where is the word already Yn-m (a-priory —) does not contain transvection species hk (*), * ^ 0, i < n - m, those. it consists entirely of diagonal letters of the alphabet (g). By

applying relations 1 and 2 to it, it is now reduced to the form ^ (sx)

o... o dn (sn) in an obvious

way, i.e. the given word is reduced to its standard form S(W). □.Completeness of relations 1-8.

Let now W=0 arbitrary group relation Tom (R) ( in generators (g)). Having written the left-

hand side in its standard form (using relations 1-8), we replace it with S(W)=0. But according to Theorem 1, the latter is possible only for zero letters of the form S(W). And this already means that the given relation W=0 can be derived from 1-8. Theorem 3 is completely proven.

As we noted above, when m=n is fn_m o... o f = 0, those. in this case, both the transvections from (g) and the (related to us) relations 3-8 disappear from our field of consideration. In other words (g) is replaced with a subalphabet dt (s), se Ro, 1 < i < n, and the relations 1 - 8 with 1.2, and here Theorem 3 simply turns into Dikov's definition of the diagonal subgroup D° (R).

4.Assignment of the projective factor PT°m (R).

Based on (main) Theorem 3, in this section we give a combinatorial representation of the group factor (R) in the centre С = centT0m (R). And to do this, we first need to calculate this center, or rather, find some C-generating system of words W of the alphabet (g). Then the factor under consideration will be presented as Km(R) = {(g)l|1 -W = 0) (см.[11], стр.77).

In the case where m=n, the group being studied T°m (R) turns into a (classical) diagonal group D°(R). Setting its projective factor is not difficult and it is not interesting.

Cases m<nB T0m (R) require additional research. Let x = (xtj ) - an arbitrary matrix from the center C. Taking also an arbitrary (diagonal) matrix d (s), s g Ro, 1 < к < n, we have

dk (s) о x = x о dk(s). The latter will obviously lead us s о хи = хи о s, those to switch on

xa g centR0. (g)

Let's consider in x its "corner" positions xij (i.e., positions for which i < n - m и j >i + m ). For these elements we also have the equalities ^ (Л) о x = x о ttj (Л)

(À-an arbitrary element from R). Comparison in last positions < i,i >, < j, j >, < j,i > will lead us to Лу = 0 = xtjÀ and

x„À = tejj-. (s) Thus, in the central matrix, all its corner elements x are required to enter the annulment

AnnR, and its diagonal elements (in addition to inclusions) must also satisfy the requirements of "scalarity" (s). Now check that the matrix x, satisfying all the above conditions, will be central in T0m (R), is no longer difficult. It also became obvious that the center C is generated by quasi-transvections ^ (â), S g AnnR (i < n - m, j > i + m, and all "scalar" words d (s ) о... о dn (sn ). Summarizing these facts, we can formulate the following result.

Theorem 4. Projective generalized m-triangular group PT0m(R), n > 2 (1 < m < n), over the associative ring R ф {o} in generators (g) is represented by relations 1-8, angular relations ty (S) = 0, S g AnnR (i < n - m, j > i + m), and with the following "scalar" relations

d1(si) о ... о dn (sn ) = 0

(s G centRO ).

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