CC BY
10
УДК 517.9
On Applications of the Cayley Graphs of some Finite Groups of Exponent Five
Alexander A. Kuznetsov* Konstantin V. SafonoV
Institute of Computer Science and Telecommunications Reshetnev Siberian State University of Science and Technology Krasnoyarsky Rabochy, 31, Krasnoyarsk, 660037
Russia
Received 17.04.2017, received in revised form 20.04.2017, accepted 16.10.2017 Let B0(2, 5) be the largest two-generator finite Burnside group of exponent five. It has the order 534. We define an automorphism ф which translates generating elements into their inverses. Let Св0(2,5)(ф) be the centralizer of ф in B0(2, 5). It is known that |Св0(2,5)(ф)| = 516. The growth functions of the centralizer are computed for some generating sets in the article. As the result we got diameters and average diameters of corresponding the Cayley graphs of Св0(2,5(ф).
Keywords: periodic group, collection process, Hall's polynomials, the Cayley graph, multiprocessor computer system.
DOI: 10.17516/1997-1397-2018-11-1-70-78.
One of the important tools for defining the structure of a group is the study of its growth with respect to a fixed generating set. Let G = (X). We call the ball Ks of radius s of a group G the set of all its elements, which can be presented as a group words with length s in the alphabet X. For each nonnegative integer s, one can define the growth function of the group F(s), which is equal to the number of elements of the group G with respect to X that can be represented as an irreducible group words with the length s. Thus,
F(0) = |Ko| = 1, F(s) = \KS\- \KS-1\ when s e N.
As a rule, the growth function of a finite group is represented in the form of a table which contains non-zero values of F(s).
Unfortunately, although the computation of the growth function of a large finite group is solvable, it is a rather complicated problem. This is due to the fact that, in general, the task of determination of the minimal word of a group element, as shown by S. Iven and O. Goldreich [1], is NP-hard. Thus, in the worst case, the number of elementary operations that must be performed to solve this problem is an exponential function of \X \.
Note also that computing the growth function of a group, we define in a parallel way the characteristics of the corresponding Cayley graph, for instance, the diameter and the average diameter [2]. Let F(s0) > 0, but F(s0 +1) = 0, then s0 will be the diameter of the Cayley graph of the group G in the generating alphabet X, which we will denote DX (G). Accordingly, the
__1 so
average diameter DX(G) is equal to -—- s ■ F(s).
^ s=0
* [email protected] [email protected] © Siberian Federal University. All rights reserved
In recent decades, the Cayley graph theory has developed as a separate large branch of the graph theory. The Cayley graphs are used both in mathematics and outside it. In particular, the Cayley graphs were used in information technology after the pioneering work of 1986 by S. Akers and B. Krishnamurti [3], who first proposed the use of these graphs to represent computer networks, including for topology modeling (i. e., methods of connecting processors to each other) multiprocessor computer systems (MCS) — supercomputers, as well as data centers [4,5]. Since then this direction is actively developing. This is due to the fact that the Cayley graphs have many attractive properties, of which we distinguish their regularity, vertex transitivity, small diameter and degree with a sufficiently large number of vertices in the graph. Note that such basic network topologies as «ring», «hypercube» and «torus» are the Cayley graphs. According to the latest data, the performance of the most powerful MCS approaches 100 petaFLOPS, with the total number of cores in the processors already exceeding 10 millions. As you know, the network topology is a critical parameter of the MCS performance. Therefore, there is a reason to believe that, not in the far future, the knowledge on very large graphs will be needed in the designing of distributed systems, in which peak performance will reach 1 exaFLOPS and above.
One of the widely used topologies of MCS is the fc-dimensional hypercube. This graph is determined by the group B(fc, 2), which is the fc-generated Burnside group of period 2. B(fc, 2), has a simple structure and is equal to the direct product of fc copies of a cyclic group of order 2. In the work [6], the Cayley graphs of the groups B(fc, 3), i. e. groups of period 3, were studied and the comparative analysis of these graphs with respect to a hypercube carried out. The analysis showed that the characteristics of B(fc, 3) are more preferable than characteristics of B(fc, 2). It means that, while paired comparison graphs B(fci, 3) and B(fc2, 2) with approximately the same number of vertices, the first ones have the smaller diameters, the average diameters and the degrees. A similar result was obtained in [7] in the study of groups of period 4. In this regard, the task of the study of the Cayley graphs of finite Burnside groups of other periods is interesting.
Let B0(2,5) = (ai ,a2) be a maximal finite two-generated Burnside group of period 5, which order is equal to 534 [8]. Using the computer algebra system GAP, it is easy to obtain pc-presentation (power commutator presentation) of this group [4,9]. In this case, each element g G B0(2, 5) can be uniquely written in the following form:
Vg G B0(2, 5) ^ g = a^ • a%2 • ... • aff, a, G Z5, i = 1, 2,. .., 34.
Here ai and a2 are the generating elements B0(2, 5), a3,..., a34 are the commutators, which are computed recursively by ai and a2 [8].
In addition to the applied interest, there is another reason for the increased attention of researchers to the growth function of B0(2,5). This is because an obtained information may be useful in solving of the open problem on finiteness of B(2, 5) which is a free two-generated Burnside groups of period 5. If B(2,5) is finite, then B0(2,5) = B(2, 5). However, in the foreseeable future, to calculate the growth function of B0(2, 5) is hardly possible, since the number of elements is very large:
534 = 582076609134674072265625 « 5 • 1023.
Note that until now, with the help of computer calculations, it was possible to obtain the growth functions of the factor-groups of the group B0(2,5), which order does not exceed 5i7 [10].
Let us consider the map v of the following form:
ai ^ a, 1
a2 ^ a2 .
It is easy to see [11], that v is an involutive automorphism of groups B(2, 5) and B0(2,5).
Let CB(2,5)(v) and CBo(2,5)(v) be the centralizers of the automorphism v in B(2, 5) and B0(2, 5) respectively. By the theorem of V. P. Shunkov [12], if CB(2,5)(v) is a finite group, then the group B(2,5) is also finite. In other words, if CB(2,5)(v) = CBo(2,5)(v), then B(2, 5) = B0(2,5). By this reason, the study of the growth of CBo(2,5)(v) is of great interest. Further, for brevity, we will write C instead of CBo(2,5)(v).
In [11], the structure of group C is studied and the following results are obtained:
1. \C| = 516,
2. C = Co x <z> where \Co\ = 515 and \<z>\ = 5;
3. z is the central element of the group B0(2, 5),
4. C0 = <X0> where \X0\ = 4 is the minimal number of the generators of C0,
5. pc-presentation of C0 is computed.
The purpose of this article is the study of the growth function of the group C with respect to the minimal generating set X = X0 U {z}, and also the symmetric — Y = X U X-1.
Since for the evaluation of growth function it is required to multiply elements of the group, then for the practical implementation an efficient algorithm for multiplication is needed. There are two ways to calculate the product of elements in the group given by pc-presentation: the collective process [4,9] and Hall's polynomials method [13]. Numerous computational experiments showed that the second method allows to multiply the elements in these groups much faster than in a collecting process (at least in exponent) [6,10,14,15].
The documentation of the computer algebra system GAP refers to the possibility of automated calculation of Hall's polynomials in the simplest cases. However, in the general case, this task is not trivial, because is not reduced to routine computation and requires the involvement of programming languages that support complex regular expressions, and also systems of computer mathematics with a wide range of procedures for symbolic computation. In fact, working with a group that has a large order, usually it is required the unique revision of the code that takes into account the feature of the group structure and characteristics of the computer.
In the following section of the article we present an algorithm to calculate the Hall's polynomials of the group C0. This algorithm in general was implemented in C++, with the exception of symbolic procedures, which were written in the MATLAB language.
In the last section, the results of computer calculations of the growth function of the group C with respect to the generating set X and Y are given.
1. The Hall's polynomials of the group C0
Let C0 = <X0> where X0 = {a,, a2, a3, a4} is the minimal generating set of C0. The following theorem is proved.
Theorem. Let aX1 ... aX55 and a,1 ... ail5 be two arbitrary elements of the group C0 recorded in the commutator form. Then their product is equal aX1 ... aX55 • ayi ... ay55 = aZ1 ... a^55 where Zi € Z5 are Hall's polynomials given by formulas (1-15).
Z10 = X10 + y 10 + 2x3y2 + 2x4yi + x7yA + 3x^ yi + x^yiy^, zii = xii + yii + 4x3 y2 + 2x4y2 + xgy4 + 3x1 y2 + x4y2y4, zi2 = xi2 + yi2 + 2x3 y2 + 2x4 y3 + xgy4 + 3x| y3 + x4y3y4,
Zi = xi + yi, Z2 = x2 + y2,
Z3 = x3 + y3, Z4 = x4 + y4,
Z5 = x5 + y5 + x2yi + 4x3y2, Z6 = x6 + y6 + x3yi + 4x3y2, Z7 = xr + yr + x4yi, Z8 = xg + ys + x4y2, Zg = xg + yg + x4y3,
(1) (2)
(3)
(4)
(5)
(6)
(7)
(8) (9)
(10) (11) (12)
Zi3 = xi3 + yi3 + 3x3y2 + 2x4yi + 2x5y4 + xry2 + 4xgyi + 2xry4 + xioy4+
9 Q 9 9 9
+ 2x4yi + x4yi + 3xry4 + 3x4yiy4 + 3x4yiy4 + 2x2x4yi + 2x2yiy4 + x4yiy2 + 4x4yiy4, (13)
Zi4 = xi4 + yi4 + 3x3y2 + 2x4y2 + 4x6y4 + 2xr y3 + 3xgyi + 2xgy4 + xiiy4 +
9 Q o 9 9
+ 2x4y2 + x4 y2 + 3xsy4 + 3x4y2y4 + 3x4'y2y4 + 4x3x4yi + 4x3yiy4 + 2x4yiy3 + 4x4y2y4, (14)
Zi5 = xi5 + yi5 + x3 y2 + 2x4y3 + xgy3 + 4xgy2 + 2xgy4 + xi2y4 + 2x| y3+
+ x3y3 + 3xgy4 + 3x4y3y1 + 3x4y3y4 + 2x3x4y2 + 2x3y2y4 + x4y2y3 + 4x4y3y4. (15)
Proof. Let's write the commutator representation of the group C0 [11]: ai, a2, a3, a4 — generators;
a5 = [a2,ai], a6j = [a3,ai], ar = [a4,ai], as = [a4,a2], ag = [a4,a3] — commutators of weight 2;
ai0 = [a4, ai, a4], aii = [a4,a2,a4], ai2 = [a4,a3, a4] — commutators of weight 3;
ai3 = [a4, ai, a4, a4], ai4 = [a4, a2, a4, a4], ai5 = [a4, a3, a4, a4] — commutators of weight 4.
List of defining relations R for commutators: (trivial relations [a,j, a»] = 1 are not given): a5 = 1(1 ^ i ^ 15), [a2, ai] = a5, [a3, ai] = a6, [a3, a2] = a| a4 a^ a4i a22 a^ a34 ai5, [a4, ai] = ar, [a4, a2] = as, [a4, a3] = ag, [a5, a4] = a23, [a6, a4] = a44, [ar, a2] = ai3, [ar, a3] = a24, [ar, a4] = ai0, [ag, ai] = a|3, [ag, a3] = ai5, [ag, a4] = au, [ag, ai] = a34, [ag, a2] = afFj, [ag, a4] = ai2, [ai0, a4] = ai3, [aii, a4] = ai4, [ai2, a4] = ai5.
Sometimes we will write g = (xi,..., xi5).
In order to determine the functions z first we need to calculate the products of ay- af for all 1 ^ i < j ^ 15, x,y = 1, 2,3,4. For the pair (j,i), it is required to find the interpolation polynomial for each of the 15 commutators by the 16 values of the product (y, x). Let's start with the first pair ay af:
a} a} = (1,1,0, 0,1,0,0,0, 0, 0,0,0,0, 0, 0
af a} = (1, 3,0, 0, 3,0,0,0, 0, 0,0,0,0, 0, 0 a2 a\ = (2,1,0, 0, 2,0,0,0, 0, 0,0,0,0, 0, 0 af a\ = (2, 3,0, 0,1,0,0,0, 0, 0,0,0,0, 0, 0 a\ af = (3,1,0, 0, 3,0,0,0, 0, 0,0,0,0, 0, 0 af af = (3, 3,0, 0, 4,0,0,0, 0, 0,0,0,0, 0, 0 a2 af = (4,1,0, 0, 4,0,0,0, 0, 0,0,0,0, 0, 0 af af = (4, 3,0, 0, 2,0,0,0, 0, 0,0,0,0, 0, 0 Let's write:
a2 a1
4 l a2 al
22 a2 al
42 a2 al
2f a2 al
4f a2 al
24 a2 al
44 a2 al
a2 al = al a2 a
l a2 af
a4
1, 2, 0, 0, 2,0,0, 0, 0,0,0,0, 0, 0,0
1,4, 0, 0,4,0,0, 0, 0,0,0,0, 0, 0,0 2, 2, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
2, 4, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
3, 2, 0, 0,1,0,0, 0, 0,0,0,0, 0, 0,0
3, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
4, 2, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 4,4, 0, 0,1,0,0, 0, 0,0,0,0, 0, 0,0
y „f — „f „y j31'2)if,v) „fi1'2)(f,y)
/i(12)(f.y)
15 ,
where /^'^(x, y) = J2P=1 J2f=1 3pqxpyq are some polynomials over the field Z5. To find them,
44 p=l q=
let's perform interpolation for each commutator r = 3,4,..., 15.
To find /r(1'2)( x, y), it is required to solve a system of linear equations over the given field:
J2J2^rpq xpyq = zyf V x,y =1, 2, 3,4,
(16)
p=} q=:L
where zyf is a value of r-th commutator for the pair (y, x). This system will have 16 variables and consist of 16 equations.
(l'2)
Let's show how to find /5 ' (x,y) at the example of the 5-th commutator. For short, we will write /pq instead of ^L. Substituting in (16) all values of zyf we receive:
111
242 343 414 224 433 132 3 2 1 334 123 422 2 3 1 44 1 1
3 2
144
111
342 243 414 243 111 414 342 342 414 111 243 414 243 342 111
1111 1342 1243 1414 2431 2222 2323
2 13 4 3421 3232 3333
3 12 4
414 431 421 4444
111 1 3 4 1 2 4 141 431 444 414 421 421 414 444 431 141 1 2 4 1 3 4 111
5
111 131 121 141 311 331 321 3 4 1 211 231 221 2 4 1 411
43 42
44
/321 3l2 3fi /22 3lf
/ 4l / f2 32f
/l4
/ 42 / ff
/24 / 4f
3f4 \3ffJ
1
2
3
4 2 4 1 3
3 1
4 2 4 3
2 1
The rank of the matrix is equal to 16, therefore it has the unique solution: ¡311 = 1, and all the remaining coefficients ftij are equal to zero. Therefore,
/5(1'2)(x,y) = xy■
In a similar way, we get that /1,2)( x, y) = 0 for all r = 5. Thus,
ay2 af = (x, y, 0, 0, xy, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). Using this method, let's calculate other noncommutative pairs ay af.
ay af = (x, 0, y, 0,0, xy, 0, 0,0,0, 0, 0, 0,0,0),
ay af = (0, x, y, 0, 4xy, 4xy, 0, 0, 0, 2xy, 4xy, 2xy, 3xy, 3xy, xy), ay4 af = (x, 0, 0, y, 0, 0, xy, 0, 0, 2xy + 3xy2, 0, 0, 2xy + 2xy2 + xy3, 0, 0), a\ af = (0, x, 0, y, 0, 0, 0, xy, 0, 0, 2xy + 3xy2, 0, 0, 2xy + 2xy2 + xy3, 0), a\ af = (0, 0, x, y, 0, 0, 0, 0, xy, 0, 0, 2xy + 3xy2, 0, 0, 2xy + 2xy2 + xy3), ay5 af = (0,0, 0, x, y, 0, 0, 0, 0,0,0, 0, 2xy, 0, 0), a6 af = (0,0, 0, x, 0, y, 0, 0, 0,0,0, 0, 0, 4xy, 0), ay7 af = (0, x, 0, 0, 0, 0, y, 0, 0, 0, 0, 0, xy, 0, 0), ay7 af = (0,0, x, 0,0,0, y, 0, 0,0,0, 0, 0, 2xy, 0), ay7 af = (0, 0, 0, x, 0, 0, y, 0, 0, xy, 0, 0, 2xy + 3x2y, 0, 0), ay af = (x, 0, 0, 0,0,0,0, y, 0,0,0, 0, 4xy, 0, 0), ay af = (0, 0, x, 0, 0, 0, 0, y, 0, 0, 0, 0, 0, 0, xy), ay af = (0, 0, 0, x, 0, 0, 0, y, 0, 0, xy, 0, 0, 2xy + 3x2y, 0),
-y ax = (x, 0, 0, 0,0,0,0, 0, y, 0,0, 0, 0, 3xy, 0),
ax2 = (0, x, 0, 0,0,0,0, 0, y, 0,0, 0, 0, 0,4xy), ay al = (0, 0, 0, x, 0, 0, 0, 0, y, 0, 0, xy, 0, 0, 2xy + 3x2y), ayw all = (0, 0, 0, x, 0, 0, 0, 0, 0, y, 0, 0, xy, 0, 0), ayn axA = (0, 0, 0, x, 0, 0, 0, 0, 0, 0, y, 0, 0, xy, 0), ay12 aH = (0,0, 0, x, 0,0,0, 0, 0,0,0, y, 0, 0, xy).
Not listed pairs are commutative, i.e. ay al
al ay.
Thus, we have a complete set of relations for the implementation of the collection process in analytical form:
y x x y fj + l (x,y) f( + 2) (x,y)
if — /-> ^ rt * rt j^1- ft
aj ai = ai aj aj+i
j+2
fr}(x,y) 15 ,
1 < i < j < 15.
(17)
Using (17) we can calculate the product aX1 ... aX55 • ayi ... ay5 = a1 ... aH5. Following this procedure, we will find all z (1-15). □
9
2. Computer calculations of the growth of the group C
The calculation of the growth function of the group C0 in alphabets X0 = {a1,a2,a3, a4} and Y0 = Xo UX-1 was carried out according to the algorithm from [10]. For efficient multiplication of elements Hall's polynomials obtained in section 2 were used. The algorithm was implemented in C++. As a tool for parallelization, it was used OpenMP library. For the calculations, it was used the computer which has 4-core processor and 32 GB of RAM, running the Linux operating system. The program was compiled by the embedded compiler GCC. Calculating growth functions for the generating set X0 takes about 1.5 hours, and for Y0 3 hours.
Then it is easy to get the growth function of the group C in alphabets generating X = {a1, a2, a3, a4, z} and Y = X U X-1. Their graphs are shown in Figs. 1,2. For clarity, an approximating Gaussian curve obtained by the method of the least squares is added on each graph.
As already mentioned, the growth function of the group that containing information about the characteristics of the corresponding Cayley graph is:
Corollary 1. DX (C) = 29; DX (C) « 21.
Corollary 2. Dy(C) = 19; DY(C) « 14.
xlO10
2.5 2
£ 15
%
o
5 i
0.5
0_
0 5 10 15 20 25 30
Length
Fig. 1. The growth function of C generated by X
xlO10
0 2 4 6 8 10 12 14 16 18 20
Length
Fig. 2. The growth function of C generated by Y
The reported study was funded by Russian Foundation for Basic Research, Government of
Krasnoyarsk Territory, Krasnoyarsk Region Science and Technology Support Fund to the research
project no. 17-47-240318.
References
[1] S.Even, O.Goldreich, The Minimum Length Generator Sequence is NP-Hard, J. of Algorithms, 2(1981), no. 3, 311-313.
[2] A.A.Kuznetsov, A.S.Kuznetsova, A parallel algorithm for study of the Cayley graphs of permutation groups, Vestnik SibGAU, 53(2014), no. 1, 34-39 (in Russian).
[3] S.Akers, B.Krishnamurthy, A group theoretic model for symmetric interconnection networks, Proceedings of the International Conference on Parallel Processing, (1986), 216-223.
[4] D.Holt, B.Eick, E.O'Brien, Handbook of computational group theory, Boca Raton, Chapman & Hall/CRC Press.
[5] M.Camelo, D.Papadimitriou, L.Fabrega, P.Vila, Efficient Routing in Data Center with Underlying Cayley Graph, Proceedings of the 5th Workshop on Complex Networks Comple Net, 2014, 189-197.
[6] A.A.Kuznetsov, The Cayley graphs of Burnside groups of exponent 3, Sib. Elektron. Math. Izv., 12(2015), 248-254 (in Russian).
[7] A.A.Kuznetsov, A.S.Kuznetsova, Perspective topologies of multiprocessor computing systems based on the Cayley graphs of groups of period 4, Vestnik SibGAU, 17(2016), no. 3, 34-39 (in Russian).
[8] G. Havas, G.Wall, J. Wamsley, The two generator restricted Burnside group of exponent five, Bull. Austral. Math. Soc., 10(1974), 459-470.
[9] C.Sims, Computation with finitely presented groups, Cambridge, Cambridge University Press, 1994.
[10] A.A.Kuznetsov, An algorithm for computation of the growth functions in finite two-generated groups of exponent 5, Prikl. Diskr. Math., 33(2016), no. 3, 116-125 (in Russian).
[11] A.A.Kuznetsov, K.A.Filippov, On an involutive automorphism of the Burnside group Bo(2, 5), Sib. Zh. Ind. Math., 13(2010), no. 3, 68-75 (in Russian).
[12] V.P.Shunkov, On periodic groups with almost regular involution, Algebra i Logika, 11(1972), no. 4, 470-494 (in Russian).
[13] Ph.Hall, Nilpotent groups, Notes of lectures given at the Canadian Mathematical Congress 1957 Summer Seminar, in The collected works of Philip Hall, Oxford, Clarendon Press, 1988, 415—462.
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О приложениях графов Кэли некоторых конечных групп периода 5
Александр А. Кузнецов Константин В. Сафонов
Институт информатики и телекоммуникаций Сибирский государственный университет науки и технологий им. М. Ф. Решетнева Красноярский
рабочий, 31, Красноярск, 660037
Россия
Пусть B0(2, 5) — максимальная конечная двупорожденная бернсайдова группа периода 5, порядок которой равен 534. Определим автоморфизм ф, который инвертирует порождающие элементы. Пусть Св0(2,б)(ф) — централизатор ф в B0(2, 5). Известно, что \C в0(2)ь)(ф)\ = 516. В настоящей работе вычислены функции роста данного централизатора для некоторых порождающих множеств. В результате были получены диаметры и средние диаметры соответствующих графов Кэли C(2,5) (ф).
Ключевые слова: периодическая группа, собирательный процесс, полиномы Холла, граф Кэли, многопроцессорная вычислительная система.
