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МАТЕМАТИКА
Вестн. Ом. ун-та. 2017. № 1. С. 15-17.
УДК 512.54 V.A. Roman'kov
A SHORT SURVEY ON PRIMITIVE SYSTEMS OF ELEMENTS IN RELATIVELY FREE GROUPS*
The main purpose of this paper is to describe some known results and outline corresponding approaches on primitive systems of elements of relatively free groups.
Key words: primitive system, relatively free groups, primitive lifting, tame system, tame automorphism.
1. Introduction
For a positive integer n and an arbitrary variety of groups S, let Gn = Fn(S) be the relatively free group of rank n, i.e., the free group of rank n in S, and let w = {w-^..., wm}, m < n, be a system of elements in Gn. The system w is said to be primitive if it can be included in some base (set of free generators) of Gn. In particular, every primitive system of n elements is a base of Gn, and vice versa.
2. Primitivity in free groups
Let Fn be the free group of rank n with base {f1t... ,fn} and let w = {w1f..., wm}, m < n, be a system of words in Fn. Primitivity of a given system w can be algo-rithmically decided (Whitehead, see Lyndon and Schupp [1, p. 30]), and there are the following nice primitivity criteria in terms of certain properties of the m x n Jacobian matrix J(w) = (dwl/dfj), over the group ring ZFn, of the Fox derivatives d/df :ZFn^ZFn, j = 1, .., n.
Theorem 1 (Birman [2] for the case m = n and Umirbaev [3] for the general case). A system of elements w = {w1( ...,wm}, m < n, is primitive in Fn if and only if there exists a matrix A e Mnxm(Z[Fn]) such that
AJ(w) = (5lJ)mxm, where Stj denotes the Kronecker delta.
3. Primitivity in free metabelian groups
Let Mn = Fn/Fn'' be the free metabelian group of rank nwith base {x1t ...,xn}, corresponding to the base {f1,.,fn}, and let An = Fi/Fi = Mn/M'n be the free abelian group of rank n with basis {a1t..., an} corresponding to the bases {f1,.,fn} and {x1,.,xn} respectively.
Let An = Z[An] be the group ring of the group An, that is the Laurent polynomial rings in the commuting variables ap, i= 1, ..., n.
By d/dxl, i = 1, ..., n, we denote the Fox partial differentiation of the group ring Z[Mn] with values in An. Then for every tuple w = {w1, ...,wm}, m < n, of elements of Mn we put J(w) = (dwl/dxj), the Jacobian matrix over An.
The following criterion for the primitivity of a system w = {w1, ...,wm}, m < n, of elements of the free metabelian group Mn has been stated and proved in the cases m = n, m = n - 1 or m < n - 2, by E. I. Timoshenko [4]. He strongly based in his proof on the famous Suslin-Quillen-Swan theorem, that cannot be applied in the case m = n - 2. The principal case m = 1, n = 3, has been solved by the author in [5]. The case m = n - 2, n > 3, has been completed by the author in [6].
Theorem 2 (Timoshenko [4], Roman'kov [5; 6]). A system of elements w = {w1,..., wm}, m < n, is primitive in Mn if and only if the ideal generated in the group ring An by all m*m minors of the Jacobian matrix J(w) is equal to the entire ring A n.
There are two other criteria for the primitivity of a system of elements of the free metabelian group that we present in the two following theorems, respectively.
Aknowledgement. The investigation was supported by RFBR, project 16-01-00577a.
© Roman'kov V.A., 2017
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V.A. Roman'kov
Theorem 3 (Roman'kov [6]). A system of elements w = {w-^... ,wm}, m < n, is primitive in Mn if and only if there is a n*m matrix A over the group ring An such that
J(w )A = {Slj)mxm.
In the case m = n this statement coincides with Bachmuth's theorem in [7].
Theorem 4 (Roman'kov [6]). A system of elements w = {w1f ...,wm}, m < n, is primitive in Mn if and only if there are Fox derivatives Dt : Z[Mn] ^ An such that Dt(Wj) = Stj for i = 1, .., n; j = 1, ..., m.
Corollary 1. The problem of determining whether an arbitrary system w = {w1,.,wm}, m < n, of elements of Mn is primitive or not is decidable.
4. Primitive lifting in relatively free metabelian groups
For a positive integer n and a variety of groups S, let Gn = Fn(S) be the relatively free group of rank n. Let fi : Fn ^ Gn be the standard epimorphism. Then every primitive system w = {w1, ..., wm}, m < n, of elements of Fn induces the primitive system fi(w) = {P(w1),.,P(wm)} of elements of Gn.
A system v = {v1,..., vm}, m < n, of elements of Gn is called tame if there is a such primitive system w = {w1,..., wm}, m < n, of elements of Fn, that induces v.
In other words, v can be lifted (via P) to a primitive system of Fn. In general, not every primitive system v is tame. We present some corresponding results below.
We say that a positive number m is the primitive rank of Gn, denoted by pr(Gn), if every primitive system of m elements of Gn is tame, and there is system of m+1 elements of Gn that is not tame, or m = n. Obviously, every primitive system of k < pr(Gn) elements of Gn is primitive. The case pr(Gn) = n means that every automorphism of Gn is tame, i.e., it is induced (via P) by some automorphism of Fn. The set TAut(Gn) of all tame automorphisms of Gn is subgroup of Aut(Gn). One can treat pr( Gn) as a measure of tameness of Aut(Gn).
The notion of primitive systemhas been introduced by Kanta and Narain Guptas with the author [8]. The main result of [8] is as follows.
Theorem 5 (Gupta C.K., Gupta N.D., Roman'kov V.A. [8]). Let Mnfi be the free metabelian nilpotent of rank n and nilpotency class c group. Then for n > 4 and m < n - 2 every primitive system v = {v1,..., vm} of elements of Mn c is tame.
Remark 1 ([8]). For each n > 2 and c > 2 there exists a primitive system v of n - 1 elements of Mn c which cannot be lifted to a primitive system of Mn. Hence, v is not tame.
The following results cover the metabelian case.
Theorem 6 (Bachmuth [7] for the case n = 2, Roman'kov [9] and independently Bachmuth and Mochizuki [10] for the case
n > 4). For n = 2 or n > 4, every automorphism of Mn is tame.
Corollary 2. For n = 2 or n > 4, one has pr(Mn) = n.
In contrast with theorem 6 the case n = 3 is exceptional.
Theorem 7 (Chein [11]). There exists a nontame automorphism of M3.
Moreover, Aut(M3) contains a lot of non-tame automorphisms. It has been shown in the following result.
Theorem 8 (Bachmuth and Mochizuki [12]). For every finite set T of automorphisms of M3 the subgroup gp(T, TAut( Gn)) is not equal to Aut(M3).
C.K. Gupta and E.I. Timoshenko [13] constructed in a relatively free metabelian group F2 (AmA), where m is a composite number, a base consisting of nontame elements. Here Am denotes the variety of abelian groups of exponent m, A = A0 denotes the variety of all abelian groups, and AmA is the product of Am by A, i.e., the variety of groups consisting of all abelian extensions of groups in Am.
Theorem 9 (Gupta and Timoshenko [13]).
1) Let r > 4; 1 < m < r - 2; p prime, k > 1. Then every primitive system w = {w1,.,wm}, m < n, of the relatively free metabelian group Fn (ApkA) is tame.
2) Let r > 2 and m is a composite number. Then there is a primitive systemw of n-1 elements in the relatively free metabelian group Fn (^mA), that is not induced by a primitive system in Mn, thus w is nontame. More detailed information on primitive systems of elements of relatively free metabelian groups of varieties AmAn can be found in [14].
To my knowledge the only known result about primitive systems of elements of arbitrary relatively free metabelian (more generally nilpo-tent-by-abelian) groups is the following statement.
Theorem 10 (Bryant, Roman'kov [15]). Let S be a subvariety of the variety NCA, consisting of all abelian extensions of nilpotent of class < c groups, where c > 1. Let n be a positive integer and write k = 2n(n+1) + 2c. Then every primitive system in the relatively free group Gn = Fn(S is induced by some primitive system of Fk, where Fn is considered as a subgroup of Fk generated by the first n free generators.
5. Relatively free groups of primitive rank zero
For n > 1, let S be a variety of groups, and let Gn = Fn (S) be the relatively free group of rank n in S. The group Gn has the primitive rank pr(Gn) zero if there is a nontame primitive element g e Gn.
Theorem 11 (Roman'kov [5]). There exists a primitive element of M3 that is not tame.
The author's proof of this theorem does not allow to present such element.
The first example of such specific element has been exhibited by Evans [16].
A short survey on primitive systems of elements in relatively free groups
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REFERENCES
[1] Lyndon R. C., Schupp P. E. Combinatorial group theory. Berlin; Heidelberg; N. Y.: Springer Verlag, 1977. 339 p.
[2] Birman J. S. An inverse function theorem for free groups // Proc. Amer. Math. Soc. 1974. Vol. 41. P. 634-638.
[3] Умирбаев У. У. Примитивные элементы свободных групп // Успехи мат. наук. 1994. Т. 49. С. 175-176.
[4] Тимошенко Е. И. О включении данных элементов в базис свободной метабелевой группы // Деп. в ВИНИТИ АН СССР. 1988. № 2699-В. 15 с.
[5] Романьков В. А. Примитивные элементы свободной группы ранга 3 // Математический сборник. 1991. Т. 182, № 7. С. 1074-1085.
[6] Романьков В. А. Критерии примитивности системы элементов свободной метабелевой группы // Украинский математический журнал. 1991. Т. 43, № 7-8. С. 996-1002.
[7] Bachmuth S. Automorphisms of free metabelian groups // Trans. Amer. Math. Soc. 1965. Vol. 118. P. 93-104.
[8] Gupta C. K., Gupta N. D., Roman'kov V. A. Primitivity in free groups and free metabelian groups // Canadian Journal of Mathematics. 1992. Vol. 44, № 3. P. 516-523.
[9] Романьков В. А. Группы автоморфизмов свободных метабелевых групп // Проблемы взаимосвязи абстрактной и прикладной алгебры : сб. науч. тр. / Академия наук СССР, Сиб. отд-ние, Вычислительный центр. Новосибирск, 1985. С. 53-58.
[10] Bachmuth S., Mochizuki H. Y. Aut(F) ^ Aut(F/ F") is surjective for free group F of rank > 4 // Trans. Amer. Math. Soc. 1985. Vol. 292. P. 81-101.
[11] Chein O. IA-automorphisms of free and free metabelian groups // Comm. Pure Appl. Math. 1968. Vol. 21. P. 605-629.
[12] Bachmuth S., Mochizuki H. Y. M-automorphisms of free metabelian group of rank 3 // J. Algebra. 1979. Vol. 55. P. 106-115.
[13] Гупта Ч. К., Тимошенко Е. И. Примитивные системы элементов в многообразиях АтАп: критерий и индуцирование // Алгебра и логика. 1999. Т. 38, № 5. С. 513-530.
[14] Тимошенко Е. И. Эндоморфизмы и универсальные теории разрешимых групп. Новосибирск: Изд-во НГТУ. 2011. 327 с.
[15] Bryant R. M., Roman'kov V. A. Automorphism groups of relatively free groups // Math. Proc. Camb. Phil. Soc. 1999. Vol. 127. P. 411-424.
[16] Evans M. J. Primitive elements in the free metabelian group of rank 3 // J. Algebra. 1999. Vol. 220. P. 475-491.
