Groups with the same lower central sequence as a relatively free group. II. Properties

Let G be a finitely generated free, free abelian of arbitrary exponent, free nilpotent, or free solvable group, or a free group in the variety AmAn, and let A = {a1,…, ar} be a basis for G. We prove that, in most cases, if S is a subset of a basis for G which may be expressed as a word in A without using elements from {al+1,…, ar} for some l < r, then S is a subset of a basis for the relatively free group on {a1,…, al}.

Download Full-text

The skeleton of a variety of groups

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700044634 ◽

1972 ◽

Vol 6 (3) ◽

pp. 357-378 ◽

Cited By ~ 6

Author(s):

R.M. Bryant ◽

L.G. Kovács

Keyword(s):

Free Group ◽

Finite Groups ◽

Abelian Groups ◽

Product Variety ◽

Finite Variety ◽

Locally Finite ◽

Variety Of Groups ◽

Locally Finite Variety ◽

Countably Infinite ◽

Relatively Free Group

The skeleton of a variety of groups is defined to be the intersection of the section closed classes of groups which generate . If m is an integer, m > 1, is the variety of all abelian groups of exponent dividing m, and , is any locally finite variety, it is shown that the skeleton of the product variety is the section closure of the class of finite monolithic groups in . In particular, S) generates . The elements of S are described more explicitly and as a consequence it is shown that S consists of all finite groups in if and only if m is a power of some prime p and the centre of the countably infinite relatively free group of , is a p–group.

Download Full-text

A solution of a problem of Plotkin and Vovsi and an application to varieties of groups

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700002056 ◽

1999 ◽

Vol 67 (3) ◽

pp. 329-355 ◽

Cited By ~ 2

Author(s):

C. K. Gupta ◽

A. N. Krasil'nikov

Keyword(s):

Free Group ◽

Invariant Subgroup ◽

Arbitrary Field ◽

Finitely Based ◽

Maximal Condition ◽

Infinite Rank ◽

Characteristic 2 ◽

Countably Infinite ◽

Fully Invariant Subgroups ◽

Relatively Free Group

AbstractLet K be an arbitrary field of characteristic 2, F a free group of countably infinite rank. We construct a finitely generated fully invariant subgroup U in F such that the relatively free group F/U satisfies the maximal condition on fully invariant subgroups but the group algebra K (F/U) does not satisfy the maximal condition on fully invariant ideals. This solves a problem posed by Plotkin and Vovsi. Using the developed techniques we also construct the first example of a non-finitely based (nilpotent of class 2)-by-(nilpotent of class 2) variety whose Abelian-by-(nilpotent of class at most 2) groups form a hereditarily finitely based subvariety.

Download Full-text

AUTOMORPHISMS OF RELATIVELY FREE GROUPS IN THE VARIETY N2A ∧ AN2 ∧ Nc

Journal of the London Mathematical Society ◽

10.1017/s0024610701002058 ◽

2001 ◽

Vol 63 (3) ◽

pp. 607-622 ◽

Cited By ~ 2

Author(s):

ATHANASSIOS I. PAPISTAS

Keyword(s):

Automorphism Group ◽

Free Group ◽

Finite Index ◽

Finite Rank ◽

Free Groups ◽

Finite Set ◽

Tame Automorphism ◽

Positive Integers ◽

Relatively Free Group

For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

Download Full-text

When endomorphisms of G inducing automorphisms of G/V are automorphisms

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500018034 ◽

1987 ◽

Vol 30 (1) ◽

pp. 115-120

Author(s):

O. Macedońska

Keyword(s):

Free Group ◽

Commutator Subgroup ◽

Vector Representation ◽

Verbal Subgroup ◽

Infinite Rank ◽

Fixed Set ◽

Free Generators ◽

Countably Infinite ◽

Relatively Free Group

Let G denote a relatively free group of a finite or countably infinite rank with a fixed set of free generators x1,x2,…,G′ the commutator subgroup, and V a verbal subgroup belonging to G′. Following H. Neumann [6] we shall use the vector representation for endomorphisms of G. Vector v = (ν1, ν2,…) represents an endomorphism v such that xiv = νi for all i. The identity map is represented by l=(x1,x2…). We need also thetrivial endomorphism 0 = (e, e,…). The length of vectors is equal to the rank of G. We shall consider the near-ring of vectors, with addition and multiplication given below u + v=(ulν1, u2ν2,…) where uiνi; is a product in G, and uv = (u1v, u2v,…) where uiv isthe image of ui, under the endomorphism v. There is only one distributivity law (u + v)w =uw + vw.

Download Full-text