scholarly journals All automorphisms of the 3-nilpotent free group of countably infinite rank can be lifted

1988 ◽  
Vol 118 (1) ◽  
pp. 120-128 ◽  
Author(s):  
Piotr Włodzimierz Gawron ◽  
Olga Macedońska
Keyword(s):  
Free Group ◽  
Infinite Rank ◽  
Countably Infinite

scholarly journals On the endomorphism near-ring of a free group

1980 ◽  
Vol 23 (1) ◽  
pp. 103-121 ◽  
Author(s):  
R. Warwick Zeamer
Keyword(s):  
Wreath Product ◽  
Free Group ◽  
The Other ◽  
Finite Support ◽  
Finite Product ◽  
Prime Factorization ◽  
Infinite Rank ◽  
One To One ◽  
Almost All ◽  
Countably Infinite

Suppose F is an additively written free group of countably infinite rank with basis T and let E = End(F). If we add endomorphisms pointwise on T and multiply them by map composition, E becomes a near-ring. In her paper “On Varieties of Groups and their Associated Near Rings” Hanna Neumann studied the sub-near-ring of E consisting of the endomorphisms of F of finite support, that is, those endomorphisms taking almost all of the elements of T to zero. She called this near-ring Φω. Now it happens that the ideals of Φω are in one to one correspondence with varieties of groups. Moreover this correspondence is a monoid isomorphism where the ideals of φω are multiplied pointwise. The aim of Neumann's paper was to use this isomorphism to show that any variety can be written uniquely as a finite product of primes, and it was in this near-ring theoretic context that this problem was first raised. She succeeded in showing that the left cancellation law holds for varieties (namely, U(V) = U′(V) implies U = U′) and that any variety can be written as a finite product of primes. The other cancellation law proved intractable. Later, unique prime factorization of varieties was proved by Neumann, Neumann and Neumann, in (7). A concise proof using these same wreath product techniques was also given in H. Neumann's book (6). These proofs, however, bear no relation to the original near-ring theoretic statement of the problem.

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

1999 ◽  
Vol 67 (3) ◽  
pp. 329-355 ◽  
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.

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

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.

scholarly journals Generating groups of certain soluble varieties

1974 ◽  
Vol 17 (2) ◽  
pp. 222-233 ◽  
Author(s):  
Narain Gupta ◽  
Frank Levin
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Free Groups ◽  
Infinite Rank ◽  
Variety Of Groups ◽  
Countably Infinite

Any variety of groups is generated by its free group of countably infinite rank. A problem that appears in various forms in Hanna Neumann's book [7] (see, for intance, sections 2.4, 2.5, 3.5, 3.6) is that of determining if a given variety B can be generated by Fk(B), one of its free groups of finite rank; and if so, if Fn(B) is residually a k-generator group for all n ≧ k. (Here, as in the sequel, all unexplained notation follows [7].)

The Abelian Case of Solitar's Conjecture on Infinite Nielsen Transformations

1985 ◽  
Vol 28 (2) ◽  
pp. 223-230 ◽  
Author(s):  
Olga Macedonska-Nosalska
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Infinite Rank ◽  
Abelian Case ◽  
Countably Infinite ◽  
Identical Transformation

AbstractThe paper proves that the group of infinite bounded Nielsen transformations is generated by elementary simultaneous Nielsen transformations modulo the subgroup of those transformations which are equivalent to the identical transformation while acting in a free abelian group. This can be formulated somewhat differently: the group of bounded automorphisms of a free abelian group of countably infinite rank is generated by the elementary simultaneous automorphisms. This proves D. Solitar's conjecture for the abelian case.

scholarly journals Powers of a product of commutators as products of squares

2004 ◽  
Vol 2004 (7) ◽  
pp. 373-375 ◽  
Author(s):  
Alireza Abdollahi
Keyword(s):  
Recent Result ◽  
Free Group ◽  
Infinite Rank ◽  
Minimum Number

We prove that for any odd integerNand any integern>0, theNth power of a product ofncommutators in a nonabelian free group of countable infinite rank can be expressed as a product of squares of2n+1elements and, for all such oddNand integersn, there are commutators for which the number2n+1of squares is the minimum number such that theNth power of its product can be written as a product of squares. This generalizes a recent result of Akhavan-Malayeri.

scholarly journals The skeleton of a variety of groups

1972 ◽  
Vol 6 (3) ◽  
pp. 357-378 ◽  
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.

COMMUTATOR SUBGROUPS OF FREE NILPOTENT GROUPS

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1021-1031
Author(s):  
N. GUPTA ◽  
I. B. S. PASSI
Keyword(s):  
Nilpotent Group ◽  
Free Group ◽  
Finite Rank ◽  
Free Groups ◽  
Nilpotent Groups ◽  
Finite Chain ◽  
Free Nilpotent Group ◽  
Infinite Rank ◽  
Derived Subgroup ◽  
Finite Set

For fixed m, n ≥ 2, we examine the structure of the nth lower central subgroup γn(F) of the free group F of rank m with respect to a certain finite chain F = F(0) > F(1) > ⋯ > F(l-1) > F(l) = {1} of free groups in which F(k) is of finite rank m(k) and is contained in the kth derived subgroup δk(F) of F. The derived subgroups δk(F/γn(F)) of the free nilpotent group F/γn(F) are isomorphic to the quotients F(k)/(F(k) ∩ γn(F)) and admit presentations of the form 〈xk,1,…,xk,m(k): γ(n)(F(k))〉, where γ(n)(F(k)), contained in γn(F), is a certain partial lower central subgroup of F(k). We give a complete description of γn(F) as a staggered product Π1 ≤ k ≤ l-1(γ〈n〉(F(k))*γ[n](F(k)))F(k+1), where γ〈n〉(F(k)) is a free factor of the derived subgroup [F(k),F(k)] of F(k) having countable infinite rank and generated by a certain set of reduced commutators of weight at least n, and γ[n](F(k)) is the subgroup generated by a certain finite set of products of non-reduced ordered commutators of weight at least n. There are some far-reaching consequences.

scholarly journals The kernel of the monodromy of the universal family of degree d smooth plane curves

2020 ◽  
pp. 1-15
Author(s):  
Reid Monroe Harris
Keyword(s):  
Plane Curve ◽  
Teichmüller Space ◽  
Mapping Class ◽  
Plane Curves ◽  
Teichmuller Space ◽  
Infinite Rank ◽  
Universal Family ◽  
Smooth Plane ◽  
Quartic Curves ◽  
Countably Infinite

We consider the parameter space [Formula: see text] of smooth plane curves of degree [Formula: see text]. The universal smooth plane curve of degree [Formula: see text] is a fiber bundle [Formula: see text] with fiber diffeomorphic to a surface [Formula: see text]. This bundle gives rise to a monodromy homomorphism [Formula: see text], where [Formula: see text] is the mapping class group of [Formula: see text]. The main result of this paper is that the kernel of [Formula: see text] is isomorphic to [Formula: see text], where [Formula: see text] is a free group of countably infinite rank. In the process of proving this theorem, we show that the complement [Formula: see text] of the hyperelliptic locus [Formula: see text] in Teichmüller space [Formula: see text] has the homotopy type of an infinite wedge of spheres. As a corollary, we obtain that the moduli space of plane quartic curves is aspherical. The proofs use results from the Weil–Petersson geometry of Teichmüller space together with results from algebraic geometry.

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