A Mal’tsev basis for a partially commutative nilpotent metabelian groupAlgebra and Logic


E. I. Timoshenko
Analysis / Logic


Algebra and Logic, Vol. 50, No. 5, November, 2011 (Russian Original Vol. 50, No. 5, September-October, 2011)




E. I. Timoshenko∗ UDC 512.5

Keywords: partially commutative nilpotent metabelian group, variety, Mal’tsev basis.

We find a canonical representation for elements of a partially commutative group in a variety of soluble groups of derived length two and nilpotency class at most c  1. 1. PRELIMINARY INFORMATION AND THE NOTATION

The objective of the paper is to find a canonical representation for elements of a partially commutative group in a variety of soluble groups of derived length two and nilpotency class at most c  1.

We start by introducing some necessary definitions and designations. As usual, for elements x and y of a group G, their commutator x−1y−1xy is denoted by [x, y]. For n  3, we put [x1, x2, . . . , xn] = [[x1, x2, . . . , xn−1], xn].

Denote by A2 a variety of all metabelian groups, i.e., all groups satisfying an identity [[x, y], [z, v]] = 1. The lower central series G = γ1(G) ≥ γ2(G) ≥ . . . of a group G is given by the rule γi+1(G) = [γi(G), G]. A variety Nc of nilpotent groups of nilpotency class at most c, c  1, consists of all groups G for which γc+1(G) = 1.

Hereinafter, Γ is a finite undirected graph without loops, whose vertex set {x1, . . . , xr} is denoted by X. If vertices xi and xj are adjacent in Γ then we write (xi, xj) ∈ Γ.

For any variety M of groups and for the graph Γ, a partially commutative group F (M,Γ) is defined as follows: the generating set of F (M,Γ) coincides with the vertex set X of Γ, and defining ∗Supported by RFBR, project No. 09-01-00099.

Novosibirsk State Technical University, pr. Marksa 20, Novosibirsk, 630092 Russia; algebra@nstu.ru.

Translated from Algebra i Logika, Vol. 50, No. 5, pp. 647-658, September-October, 2011. Original article submitted December 3, 2010; revised February 25, 2011. 0002-5232/11/5005-0439 c© 2011 Springer Science+Business Media, Inc. 439 relations are of the form xixj = xjxi if xi and xj adjacent vertices of Γ. In addition, the group

F (M,Γ) belongs to the variety M. Thus F (M,Γ) is represented as

F (M,Γ) = 〈X | xixj = xjxi ⇐⇒ (xi, xj) ∈ Γ; M〉 (1) in M. The graph Γ is said to be defining for the group F (M,Γ). For convenience, we denote the group F (A2,Γ) by SΓ, and the F (Nc ∧ A2,Γ) by GΓ,c.

Having a convenient canonical representation of elements is useful in studying properties of a group. A representation of elements for a partially commutative group F (M,Γ) defined by representation (1) in the variety of all groups was specified in [1], where it is underpinned by the idea of expressing elements of the group in terms of a product of mutually commuting blocks.

A handy canonical representation for suitable degrees of elements in the commutator subgroup of a partially commutative metabelian group SΓ can be found in [2], in which elements of the ring

Z(SΓ/S′Γ) are treated as degree exponents. The representation in [2] made it possible to obtain a number of helpful properties for SΓ and its universal theory.

For elements of a torsion-free finitely generated nilpotent group, a canonical representation derives by reason of the fact that such a group has a Mal’tsev basis. Recall the definition of a

Mal’tsev basis.

Let G be a torsion-free finitely generated nilpotent group. We know that G has a central series of the form

G = G1 > G2 > . . . > Gs+1 = 1 with infinite cyclic factors (see [3]). Take elements a1, . . . , as satisfying Gi = gp〈ai, Gi+1〉. An ordered system {a1, . . . , as} of elements is called a Mal’tsev basis for G. Every element g ∈ G is uniquely represented as g = at11 . . . a ts s , ti ∈ Z.

We prove that a partially commutative group GΓ,c has a Mal’tsev basis, which can be obtained by refining the lower central series of GΓ,c. This is equivalent to being torsion free for factors in the lower central series of GΓ,c.

Despite the fact that partially commutative metabelian groups SΓ, being approximated by torsion-free nilpotent groups, do not contain elements of finite order [2], it is not obvious that GΓ,c lacks elements of finite order. In fact, it is easy to point out a torsion-free metabelian group G for which the quotient G/γ2(G) contains elements of finite order. Such is, for instance, a metabelian group generated by two elements {x, y} and defined by one relation x2[x, y] = 1.

It is well known that for any group G and for elements u1, u2 ∈ γn(G) and v ∈ γm(G), the following congruence holds: [u1u2, v] ≡ [u1, v][u2, v] (modγn+m+1(G)) (2) (see, e.g., [4]). 440

Denote by Gc a free group in the variety Nc ∧ A2, and by X = {x1, . . . , xr} its basis. For any c  1, a set Bc of commutators is defined by induction. Put B1 = X. For c  2, the set Bc consists of all commutators w of weight c having the form w = [xi, xj , xj1, . . . , xjc−2 ], (3) where 1  j < i  r, j  j1  . . .  jc−2.

PROPOSITION 1 [5]. For c  1, elements of Bc constitute a basis for a free Abelian group γc(Gc).

On a set X, we introduce the following order: x1 < x2 < . . . < xr.

Denote by w(n) the nth letter in a commutator of form (3), with 1  n  c. The order above is extended to a set B = ∞⋃ c=1

Bc of commutators as follows: (1) for u, v ∈ Bc, put u > v if u(1) = v(1), . . . , u(n − 1) = v(n− 1), with u(n) > v(n); (2) for u ∈ Bp and v ∈ Bq, put u > v whenever p > q.

Proposition 1 gives rise to a known result on a canonical representation of elements for the group Gc.

PROPOSITION 2. The set c⋃ m=1

Bm on which the order is defined as above is a Mal’tsev basis for Gc obtained by refining the lower central series of Gc. 2. CANONICAL REPRESENTATION OF ELEMENTS FOR A PARTIALLY


Let M be some variety of groups. It is known that for any groups Gλ, λ ∈ Λ, M contains a group G, which is called an M-product, or verbal product, of Gλ. An M-product G of groups Gλ is defined by setting