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йй! 10.25513/1812-3996.2018.23(2).27-34 АЛГОРИТМЫ ДЛЯ МЕТАБЕЛЕВЫХ ГРУПП
(посвящается 70-летию профессора Виталия Анатольевича Романькова)
А. В. Меньшов12, А. Г. Мясников1, А. В. Ушаков1
1Институт технологий Стивенса, Хобокен, Нью-Джерси, США
2Омский государственный университет им. Ф. М. Достоевского, г. Омск, Россия
Дата принятия в печать 29.03.2018
Дата онлайн-размещения 25.06.2018
Ключевые слова
Метабелевы группы, проблема равенства, проблема сопряженности, алгоритм, коммутативная алгебра, базисы Гребнера
Финансирование
Исследование первого автора выполнено при финансовой поддержке РНФ в рамках научного проекта № 16-11-10002
ALGORITHMS FOR METABELIAN GROUPS
(paper dedicated to Professor Vitaly Anatol'evich Roman'kov on the occasion of his 70th birthday)
А. V. Menshov12, A. G. Myasnikov1, A. V. Ushakov1
1Stevens Institute of Technology, Hoboken, NJ, USA 2Dostoevsky Omsk State University, Omsk, Russia
Article info Abstract. In this paper, we begin the study of computational complexity of the principal
Received algorithmic problems in finitely generated metabelian groups. The main goal is to classify
11.03.2018 the algorithmic problems in metableian groups in terms of their computational complexity.
Accepted 29.03.2018
Available online 25.06.2018
Keywords
Metabelian groups, word problem, power problem, conjugacy problem, algorithm, commutative algebra, Groebner bases
Информация о статье
Дата поступления 11.03.2018
Аннотация. Данной статьей мы начинаем систематическое изучение трудоемкости основных алгоритмических проблем в конечно порожденных метабелевых группах. Основной целью этой работы является классификация алгоритмических проблем в метабелевых группах в соответствии с их вычислительной сложностью.
Вестник Омского университета 2018. Т. 23, № 2. С. 27-34
-ISSN 1812-3996
Acknowledgements
The reported study of the first author was funded by RSF to the research project № 16-11-10002
1. Introduction
In this paper we begin the study of computational complexity of the principal algorithmic problems in finitely generated metabelian groups. Our approach here is two-fold: firstly, we rewrite and streamline some classical algorithms in metabelian groups to fit them into the framework of Groebner bases and commutative algebra (sometimes this requires a significant rebuild); secondly, we show that in most cases this reduction to the Groebner bases is in polynomial time. The main goal for the subsequent papers is to classify the algorithmic problems in metableian groups in terms of the logspace and circuit complexities.
In section 2 we introduce necessary definitions and results related to Groebner bases and state Theorem 2.1 and Corollary 2.2 that allow us to compute module presentation of ideals in polynomial rings.
In section 3 we discuss presentation of group rings of finitely generated abelian groups and modules over such rings by polynomials.
In section 4 we interpret submodule computabil-ity in terms of Groebner bases.
In section 5 we interpret word, power, and conju-gacy problems in finitely generated metabelian groups in terms of Groebner bases.
2. Groebner bases
In this section we will introduce some necessary definitions and results related to Groebner bases. For a detailed exposition we refer to [1] or [2].
Let R be a commutative ring with 1, X = [x1,... ,xn} be a finite set of variables, and = R[x1,...,xn]. A term t in the variables x1,...,xn is a power product of the form x^1 ■ ... ■ x1^ with et £ N for 1 < i <n. In particular, 1 = x10 ■ ...■ x0 is a term. We denote by T(X) the set of all terms in these variables. The divisibility relation | on T(X) is defined by sit iff there exists s' £ T(X) such that ss' = t.
A term order < is a linear order on T(X) that satisfies the following conditions:
1. 1< t for all t £ T(X).
2. t1 < t2 implies t1s < t2s for all s, t1, t2 £ T(X).
A monomial in the variables [x1, ...,xn} over R is
a polynomial of the form m = at with 0 ± a £ R and t £ T(X). Here, a is called the coefficient of m and t the term of m. The set of all monomials (in variables
[x1,..., xn} over R) is denoted by M(X,R). Multiplication on M(X,R) is defined by a1t1^a2t2 = (a1a2)(t1t2), and M(X,R) is clearly a commutative monoid.
For a term order < we define the relation ^ on M(X,R) by setting
as ^ bt iff s < t for 0±a,b £R and s,t £ T(X). We will call ^ the quasi-order (reflexive and transitive relation) on M(X,R) induced by <. If m1,m2 are two monomials with the same term but with different coefficients, then m1 ± m2 but m1 ^ m2 and m2 ^ m1. In this case we will say that m1 and m2 are equivalent w.r.t. ^ and write m1 ~ m2. Further we will denote this induced relation ^ by <.
Clearly, every polynomial f £ R[X] has a unique representation in the form Y,'i=1mi with m^ £ M(X,R) and m1 > ••• > mk. The set of monomials occurring in such representation is denoted by M(f) and called the set of monomials of /. The set T(f) of terms of f is the set of all terms of monomials m £ M(f). The set C(f) of all coefficients of f is the set of all coefficients of monomials m £ M(f).
For any finite, non-empty subset A of M(X,R) consisting of pairwise inequivalent monomials, we define max(4) to be the unique maximal element of A w.r.t. <. For any non-zero polynomial f £ R[X] we define w.r.t. < the head term HT(/) = max(T(f)), the head monomial HM(/) = max(M(f)), and the head coefficient HC(f) to be the coefficient of HM(/). The reductum red(f) of f w.r.t. < is defined as f — HM(/), i.e., f = HM(/) + red(f). A polynomial f £ R[X] is called monic w.r.t. < if f ± 0 and HC(/) = 1.
For the rest of this section, let R be a PID (or just 1).
Let m1 = a1t1 and m2 = a2t2 be monomials in fl[X]. We say that m2 divides m1 and write m2im1 if there is a monomial m3 £ such that m1 = m2m3.
Let f,g,p £ with f,p±0, and let P be a subset of fl[X]. Then we say that
1. f D-reduces to g modulo p by eliminating m
(notation f ^ g[m]), if m£M(f) is such that
p
HM(p)|m, say m = m' ■ HM(p), and g = f — m'p.
2. f D-reduces to g modulo p (notation f ^ g), if
p
f ^ g[m] for some m £ M(f).
p
3. f D-reduces to g modulo P (notation f — g), if
f — g for some p E P.
p
4. f is D-reducible modulo p if there exists g E
P[X] such that f — g.
p
5. f is D-reducible modulo P if there exists g E P[X] such that f — g.
If f is not D-reducible modulo p (modulo P), then we say f is in D-normal form modulo p (modulo P). A D-normal form of f modulo P is a polynomial g that is in D-normal form modulo P and satisfies f—t9
where — is the reflexive-transitive closure of —. We
p p
call f — g[m] a top-D-reduction of f if m = HM(/).
p
Whenever a top-D-reduction of f exists (with p E P), we say that f is top-D-reducible modulo p (modulo P).
Let 0 ± f E R[X]. A standard representation of f w.r.t. a finite subset P of P[X] is a representation
f = ^mi Pi'
™-i Pi,
with monomials mt and pL £ P such that HT(miPi) < < HT(f) for 1<i<k.
Lemma ([1], Lemma 10.3): Let P be a finite subset of R [X], 0± f £R[X], and assume that f — 0. Then
f has a standard representation w.r.t. P.
Definition (D-Grobner basis, [1], Definition 10.4) A D-Grobner basis is a finite subset G of P[X] with the property that all D-normal forms modulo G of elements of Id(G) equal zero. If I is an ideal of P[X], then a D-Grobner basis of I is a D-Grobner basis that generates the ideal /.
In other words, G is a D-Grobner basis if f — 0
' G
for every f £ Id(G).
Theorem 10.14 of [1] provides an algorithm which, when given a finite subset P of fl^], finds a D-Grobner basis G such that Id(P) = Id(C).
Unfortunately, having a D-Grobner basis G, — will
not give us unique normal forms, which means that f + Id(G) = h + Id(G) will not imply f — q and
G
h—*q. For example, consider the ring Z[x] and
G
G = {2x + 1}, then f(x) = 2x2 + 2x has the two normal forms hi= x and h2 = —x — 1.
However, for Euclidean domains with unique remainders (in the sense of [1, Definition 10.16]) the theory can be improved so that we obtain unique normal forms. We note that examples of such domains are TL and K[X] for any field K.
Now we define a new type of reduction over Euclidean domain with unique reminders.
Definition (E-reduction, [1], Definition 10.18) Let R be a Euclidean domain with unique reminders and f,g,pER[X]. We say that f E-reduces to g
modulo p and write f ^ g if there exists a monomial
p
m = atE M(f) such that HT(p)|t, say t = sHT(p), and
g = f- qsp,
where 0 Ф q E R is the quotient of a upon division with unique reminder by HC(p).
E-reduction modulo a finite subset of ДСТ, E-re-ducibility, and E-normal forms are defined in the obvious way. It is clear that E-reduction extends D-reduc-tion, i.e., every D-reduction step is an E-reduction step.
To obtain the desired bases that allow the computation of unique normal forms, we do not need another Grobner basis algorithm. It will suffice to take a D-Grob-ner basis G and E-reduce modulo G [1, Theorem 10.23].
Further, when we refer to Grobner bases and reductions, we will assume a D-Grobner bases and E-re-ductions. In the reset of the section we use the theory above to obtain results required for algorithms in metabelian groups.
Let P = {fu-Jq] be a finite subset of R[X], where R is a PID. Ideal
Id(P) = {f1a1 + - + fqaq\aiER[X]] may be treated as the fi[X]-module generated by ft.-.fq, then \d(P) = F/N, where F is the free fi[X]-module generated by and N is a sub-
module of F. Since ДЭД is Noetherian, so is F, therefore N is finitely generated, and Id(P) is finitely presented as an fi[X]-module. Our purpose will be to find its presentation.
Observe that the set S = {(a1, ...,aq) \ at E R [X]] of all solutions of the equation fiK + - + fqhq = 0 with indeterminates h1,.,hq is an R [X]-submodule of R[X]4. The set S is called the (first) module of syzygies of (f1, —,fq). Computing a finite set of generators for S is a well known problem, and its solution actually gives us an fi[X]-module presentation of Id(P). Proposition 6.1 of [1] solves this problem for the case when R is a field. Below we state analogous results that allows us to compute a presentation of Id(P) for the case when R is a PID.
Theorem 2.1 Let G be a Grobner basis. Then there exists an algorithm to find a finite presentation of R [X]-module Id(G) in terms of elements of G.
Corollary 2.2 Let P be a finite subset of P[X]. Then there exists an algorithm to find a finite presentation of R [X]-module Id(P) in terms of elements of P.
3. Representing group rings and modules by polynomials
Let A be a finitely generated abelian group given by its abelian presentation
A = (%1,...,xn | T"1(X1,...,xn),... ,rs(X1,...,xn)),
where ri(x1, ...,xn) = x±11 ■ ... ■ xnin, Cy £ 1.
For any a = x^1... x£ A we denote by T(a) the term in variables x1,y1,...,xn,yn of the form T(a) = s|fcl1 ...s^, where Si = Xi if kt > 0 and s^ = yt otherwise. Observe that the inverse procedure is obvious. Take
Ra = 1[x1,y1,...,xn,yn],
Pa = [x1y1 — 1, -,xnyn
—1,T(r1)—1.....T(VS) — 1}^RA,
and consider the map
t1:za^ra№pa), defined for a = Yl1i=1 k at £ 1A by
(a) = ^kiT(ai) + Id(PA).
=1
Clearly, 1 is a ring automorphism.
Remark 3.1 In practice, the element a would be presented as an element of Laurent ring with integer coefficients in variables x1,..., xn. Although a may have different such presentations if A is not free, given a particular presentation, we uniquely pick the polynomial Y,7i=1kiT(ai) as a representative of the residue class T1(a). For convenience, we further denote this representative by p(r1(a)). Inversely, given any representative g of the residue class T1(a) as a polynomial in Ra, we uniquely pick the corresponding element ft of Laurent ring with integer coefficients in variables x1, ...,xn and interpret it as an element of 1A and as a preimage of g, so that T1(fi) = g + Id(PA).
Further, when we refer to the size of a, we will assume the size of 1( a), which is actually the size of the polynomial p(r1(a)).
Let A be a finitely generated abelian group and t: 1A ^ RT/Id(PT) be a ring isomorphism, where RT is a ring of polynomials with integer coefficients and Pr c Rr is finite.
Let F be a free right 1i4-module with basis ..., , then any f £ F can be written uniquely in the form
f = $1a1 + - + $qaq,(ai£lA).
This form can be naturally viewed as a polynomial in ^ with coefficients in 1A. Consider the ring 1A[^1,.,^q] and its ideal Id(P,), where P, = [tej I 1 < i < j < q}, and observe that the factor ring 1A[..,^q]/Id(P,) is a free 14-module with basis [1,^1,.,^q}. Hence the natural map F ^ 1A[Z,r,... ,%q]/Id(P,) is a 1i4-module monomor-phism. Denote
RF=RA$1.....fq],
PF = PTUP,C Rf,
then
RF/Id(PF) - RA$1.....$q]/Id(PF) -
- (RJId(PTm1.....tq]/Id(P0 -
- .....$q]/Id(P0.
So we define the embedding
9T:F^RF/Id(PF), that maps an element f of the defined above to
q
eT(f) = ^ir(ai) + Id(P,) =
i=1
n
=1
Шт(«д) + ЩРР).
We define the size of f w.r.t. 9r as the sum of sizes of T(a1),...,T( aq).
Arguments of Remark 3.1 apply to representation of f as well. In the same way, by p(dr(f)) £ RF we denote the representative of the residue class dr(f).
4. Submodule computability
All modules under consideration will be right modules. Let R be a ring. If M is an P-module generated by a1,..., aq, then we write
M = modfi (a1,..., aq).
If F is a free P-module with basis (1,..., (q, then any f £ F can be written uniquely in the form f = f1r1 + -+fqrq,(n£R).
Let (:F ^ M be an P-module epimorphism define by ) = at, i = 1,... ,q. If K = kercp is the submodule of F generated by the words
№1(^1.....{q).....Wp^1.....fq)},
where the Wi((1, ...,(q) are given explicitly as words in (1,..., (q, then we write
M = (a.1,...,a.q I w1(a.1,...,aq),...,wp(a.1,...,aq)) for the corresponding presentation of M. If R is right No-etherian, then any finitely generated P-module M has a finite presentation where the number of relations is finite. By Hall's results [3] this is the case for R = 1G where G is a polycyclic-by-finite group, and, in particular, for R = 1A where A is a finitely generated abelian group.
m
=1
1
If M is presented as above, then a word of M is an fi-linear combination of the form a1r1 + —+ aqrq, (r¿ E R). Membership in a submodule L of M is decida-ble if there is an algorithm which determines for any word w of M whether or not w belongs to L (i.e. represents an element of L).
If R is a right Noetherian ring, then any finitely generated fi-module M is finitely presented and submodules of M are always finitely generated, hence finitely presented.
Definition 4.1 An fi-module M over a right Noetherian ring R is called submodule computable if for any finite set [v1, ...,vn} of words of M there is
1. an algorithm to compute a finite presentation of the submodule L of M generated by [v1,..., vn} on the given generators.
2. an algorithm to decide membership in L.
Definition 4.2 A right Noetherian ring R is called
submodule computable if finitely presented fi-modules M are submodule computable uniformly in the presentation for M.
One of the principal results of [4] is the following theorem.
Theorem 4.3 ([4], Theorem 2.12) Integral group ring of a polycyclic-by-finite group is submodule computable.
In particular, integral group ring of a finitely generated abelian group is submodule computable, so for any finitely generated metabelian group G its derived subgroup G' is submodule computable as a ZGab-mod-ule.
In the rest of this section we provide a practical proof in terms of Groebner bases for Theorem 4.3 for the case of integral group rings of finitely generated abelian groups.
Let A be a finitely generated abelian group, M be a finitely presented Z.4-module given by its presentation, and F be a free Z^-module on ,
p
so M — — where K is the submodule of F generated by
W1($1,—,$q),—,Wp($1,—,$q). Let Vl(al,—,aq),—, vn(a1,—,aq) EM and L is the Z^-submodule of M generated by these words. Denote by N the full preimage of L under the natural homomorphism F ^ F/K, clearly it is a submodule of F generated by the words
{Wl($l, — ,$q),—,Wp($l,—,$q),
Mil.....U.....vn((l.....Sq)}.
and L — N/K.
A word w(a1, — , aq) E M belongs to L iff w(i;1,—,^q) belongs to N. In particular,
wia-t, ..., aq) = 0 in M iff ..., belongs to the submodule K of F. So it is sufficient to decide membership for finitely generated submodules of free ZA-modules.
Analogously, if N has a Z4-module presentation N = (wi,..., wp, Vi,..., vn | Zi,..., zt), where zt are words in the given generators, then
L - (vi,...,vn I zi',...,zt'), where z/ is obtained from zt by replacing Wj with 0. So it is sufficient to compute presentations of finitely generated submodules of free Z4-modules.
Suppose that N is a submodule of F generated by ui,...,un, where ui = T1qk=i^kaik, aik E ZA, and i = 1, ...,n. We map elements of F to RF/Id(PF) using 8T as defined in section 3.
Lemma 4.4 Let w E F, then w E N iff 8r(w) belongs to the ideal I of RF/Id(PF) generated by eT(ui),...,eT(un).
Corollary 4.5 Let wEF, then w EN iff p(8T(w)) belongs to the ideal ] of RP generated by pFv[p(eT(ui))n = i.....n}.
These results reduce submodule membership problem for F to ideal membership problem for polynomial ring Rf over integers, which can be solved using Grobner bases technics, see [1], [2].
Lemma 4.6 Submodule N of F and ideal I of RF/Id(PF) generated by 9T(ui), ...,9T(un) are isomorphic as Z^-modules. Given a Z^-module presentation of I, one can compute the corresponding Z^-module presentation of N.
Corollary 4.7 Given an RF-module presentation of the ideal J of RP generated by PF U [p(9T(ui)) I i = 1,...,n}, one can compute the corresponding Z^-module presentation of /.
So computation of submodules' presentations in F reduces to computation of ideals' presentations in polynomial ring RF over integers, which can be done by Corollary 2.2.
5. Algorithmic problems in metabelian groups
Denote by <A2 the variety of all metabelian groups. It is known that finitely generated metabelian groups are finitely presented in <A2, which means that any finitely generated metabelian group G has a presentation of the form
G = (Xi,X2, ...,xn 1 ri, ...,rp)^2,
meaning that G - Mn/N, where Mn = Fn/F() is the free metabelian group of rank n, Fn is the free group of rank n, and N is the normal closure of ri,..., rp in Mn. The presentation above is called ^-presentation or metabelian presentation of G.
Ы
A metabelian group G is an extension of abelian normal subgroup G' by abelian group Gab = G/G'. The group Gab acts on G' by conjugation b(aG') = ba, where b £ G' and a £ Gab. This action naturally extends to the action of the group ring ЪGab on G':
( n \ n
^ka) = n(bai)ki. \i=i ) i=i
Thus derived subgroup G' of a finitely generated metabelian group G is a module over the finitely generated commutative ring Ъ Gab.
For algorithmic problems, it is advantageous to work with special Л2-presentations. For our convenience, we slightly alter the original definition from [5].
Definition 5.1 (Preferred presentation) By a preferred presentation of a finitely generated metabelian group G we mean a finite ^-presentation of the form G = (x±,x2,...,xn \ R1UR2),A2,
where:
1. R is a finite set of words of the form
[Xi,Xj]xk 1[Xj,Xk]xl 1[xk,Xi]XJ
Xj—1 _
1,
П [Xj,Xt]aiJ,
{( i.j )|1< i<j<n}
where ai} £ 1Gab.
2. R2 is a finite set of words rt of the form
mt1 min
Хг • ... • Xn W,
where тц £ Ъ, w is a word of the form that elements of R1 have, and the matrix M = (т¿j) is full rank.
So the words in R2 determine a finite presentation of the group Gab, while those in R1, as we will show later, form a part of relations for a finite Ъ Gab-presentation of G' in the generators [xj, xi] = x]~1x-1xjxi, l<i<j<n.
Below we state Theorem 9.5.1 of [6] for our version of the definition of preferred presentation.
Theorem 5.2 There is an algorithm which, when given a finitely generated metabelian group G by its finite Л2-presentation, finds a preferred ^2-presenta-tion of G.
Let G be a metabelian group generated by x1,...,xn. Its derived subgroup G' is a %Gab-module generated by [xj,xt]. Since the ring ZGab is Noetherian, G' is finitely presented as a %Gab-module. Thus a finite description of G' exists, even if it is not finitely generated as a group.
The module Mn' of the free metabelian group Mn with basis {x1, ..., xn} for n = 2 is free over ЪА2 (where A2 is the free abelian group of rank 2) with generating element [x2,x1]. For n > 3 the module Mn' is not free. All relations in the generators [Xj,xi], l < i < j < n, follow from Jacobi relations
for i,j,k = l, ...,n.
The following result is fundamental and admits an effective proof.
Theorem 5.3 ([5], Theorem 3.1) There is an algorithm which, when given a finitely generated metabelian group G by its finite ^-presentation, finds a finite ЪGab-presentation of G'.
In the rest of the section we review some classical algorithmic problems, namely, word, power, and conju-gacy problems, that have been studied earlier in [5], [7], [8], [9]. We show that all these problems can be interpreted in a unified way in terms of Groebner bases. We note that conjugacy problem, in addition to Groebner bases, requires additional tool called Noskov's Lemma.
Word problem. Solvability of the word problem in finitely generated metabelian group G may be proved by observing that G is residually finite and finitely presented in the variety Л2, so the standard procedure enumerating all finite quotients of G and consequences of defining relations in G solves the problem.
A simpler solution of the word problem was later provided by Timoshenko in [8].
Having all the machinery introduced above, it is now easy to reduce the word problem in a finitely generated metabelian group G to the ideal membership problem in a multivariate polynomial ring over integers.
Suppose that G is given by its metabelian presentation and w £ G. For any g £ G we denote g = gG' £ Gab. To check whether or not w = l in G perform the following steps:
1. Compute a preferred presentation of G using Theorem 5.2.
2. Check if W = l in Gab. It is possible since relations from R2 give us a presentation of Gab. If W Ф l, then w Ф l in G,so further we assume W = l.
3. Rewrite w as an element of G', i.e. in the form W = n[Xj,Xi]a4, where atj £ ЪGab.
4. Compute ЪGab-presentation of G' using Theorem 5.3
G' = (fa ll<i<j<n}l W1(^).....Wptfji)).
5. Using Corollary 4.7, check if in the free ЪGab -module F generated by ^ the word wtfj^ belongs to the submodule generated by w1(^, ..,wp(^ji).
Power problem. By the power problem in a group G we mean the problem of deciding for given u,v £ G whether or not v = uk for some k £ Ъ.
For a finitely generated metabelian group G, we first consider this problem for elements of G'.
Вестник Омского университета 2018. Т. 23, № 2. С. 27-34
ISSN 1812-3996-
Using the fact that we can get normal forms modulo an ideal in a polynomial ring over integers, one proves the following
Lemma 5.4 There is an algorithm which, when given a finitely generated abelian group Q, a finitely generated ZQ-module M, and elements a,b EM, decides if there exists к EZ such that b = ak.
Then the general case can be reduced to Lemma 5.4. Theorem 5.5 There is an algorithm which, when given a finitely generated metabelian group G by its finite ^-presentation and elements u,v E G, decides if there exists к EZ such that v = uk.
Conjugacy problem. The conjugacy problem in finitely generated metabelian groups was solved by Nos-kov [9]. The proof utilizes the following algorithm for rings.
Lemma 5.6 (Noskov's Lemma) There is an algorithms which, when given a finitely generated commutative ring R and a finite subset X of the group of units U(R), finds a finite presentation of the subgroup (X).
As with power problem, the proof consists of two steps, where the first one requires Noskov's lemma.
Lemma 5.7 ([5], Lemma 3.7) There is an algorithm which, when given a finitely generated abelian group Q, a finitely generated ZQ-module M, and elements a,b E M, decides if a and b are ^-conjugate, i.e. if there exists q E Q such that b = aq.
From the lemma above, the general case follows: Theorem 5.8 ([5], Theorem 2.3) There is an algorithms which, when given a finitely generated metabelian group G and elements x,y E G, decides if x and у are conjugate in G.
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ИНФОРМАЦИЯ ОБ АВТОРАХ
Меньшов Антон Владимирович - кандидат физико-математических наук, сотрудник докторантуры отделения математических наук, Институт технологий Стивенса, Хобокен, Нью-Джерси, США; ИМИТ (Институт математики и информационных технологий), Омский государственный университет им. Ф.М. Достоевского, 644077, Россия, г. Омск, пр. Мира, 55а; e-mail: [email protected], [email protected].
INFORMATION ABOUT THE AUTHORS
Menshov Anton Vladimirovich - Candidate of Physical and Mathematical Sciences, Postdoctoral Fellow, the Department of Mathematical Sciences, Stevens Institute of Technology, 1, Castle Point Terrace, Hoboken, NJ, 07030, USA; IMIT (the Institute of Mathematics and Information Technologies), Dostoevsky Omsk State University, 55a, pr. Mira, Omsk, 644077, Russia; e-mail: [email protected], [email protected].
Мясников Алексей Георгиевич - доктор физико-математических наук, профессор, руководитель отделения математических наук, Институт технологий Стивенса, Хобокен, Нью-Джерси, США; e-mail: [email protected].
Myasnikov Alexei Georgievich - Doctor of Physical and Mathematical Sciences, Professor, Department Chair, the Department of Mathematical Sciences, Stevens Institute of Technology, 1, Castle Point Terrace, Hoboken, NJ, 07030, USA; e-mail: [email protected].
Ушаков Александр Владимирович - доктор философии, профессор отделения математических наук Институт технологий Стивенса, Хобокен, Нью-Джерси, США; e-mail: [email protected].
Ushakov Alexander Vladimirovich - Doctor of Philosophy, Associate Professor, the Department of Mathematical Sciences, Stevens Institute of Technology, 1, Castle Point Terrace, Hoboken, NJ, 07030, USA; e-mail: [email protected].
ДЛЯ ЦИТИРОВАНИЯ
Меньшов А. В., Мясников А. Г., Ушаков А. В. Алгоритмы для метабелевых групп // Вестн. Ом. ун-та. 2018. Т. 23, № 2. С. 27-34. ЭО!: 10.25513/1812-3996.2018.23(2).27-34. (На англ. яз.).
FOR QTATIONS
Menshov A.V., Myasnikov A.G., Ushakov A.V. Algorithms for metabelian groups. Vestnik Omskogo univer-siteta = Herald of Omsk University, 2018, vol. 23, no. 2, pp. 27-34. DOI: 10.25513/1812-3996.2018.23(2).27-34.
