Generators of the automorphism group of a free nilpotent group

1976 ◽  
Vol 15 (4) ◽  
pp. 289-292 ◽  
Author(s):  
A. V. Goryaga
Keyword(s):  
Automorphism Group ◽  
Nilpotent Group ◽  
Free Nilpotent Group

An independence theorem for automorphisms of torsion-free groups

1981 ◽  
Vol 90 (3) ◽  
pp. 403-409
Author(s):  
U. H. M. Webb
Keyword(s):  
Automorphism Group ◽  
Nilpotent Group ◽  
Free Group ◽  
Automorphism Groups ◽  
Torsion Free Group ◽  
Free Nilpotent Group ◽  
Free Abelian Groups ◽  
Independence Theorem ◽  
The Relationship ◽  
Torsion Free

This paper considers the relationship between the automorphism group of a torsion-free nilpotent group and the automorphism groups of its subgroups and factor groups. If G2 is the derived group of the group G let Aut (G, G2) be the group of automorphisms of G which induce the identity on G/G2, and if B is a subgroup of Aut G let B¯ be the image of B in Aut G/Aut (G, G2). A p–group or torsion-free group G is said to be special if G2 coincides with Z(G), the centre of G, and G/G2 and G2 are both elementary abelian p–groups or free abelian groups.

Automorphism sequences of free nilpotent groups of class two

1976 ◽  
Vol 79 (2) ◽  
pp. 271-279 ◽  
Author(s):  
Joan L. Dyer ◽  
Edward Formanek
Keyword(s):  
Automorphism Group ◽  
Nilpotent Group ◽  
Finite Rank ◽  
Nilpotent Groups ◽  
Split Extension ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Image Position

In this paper we prove that the automorphism group A(N) of a free nilpotent group N of class 2 and finite rank n is complete, except when n is 1 or 3. Equivalently, the centre of A(N) is trivial and every automorphism of A(N) is inner, provided n ≠ 1 or 3. When n = 3, A(N) has an our automorphism of order 2, so A(A(N)) is a split extension of A(N) by . In this case, A(A(N)) is complete. These results provide some evidence supporting a conjecture of Gilbert Baumslag that the sequencebecomes periodic if N is a finitely generated nilpotent group.

scholarly journals DISTORTION IN FREE NILPOTENT GROUPS

2010 ◽  
Vol 20 (05) ◽  
pp. 661-669 ◽  
Author(s):  
TARA C. DAVIS
Keyword(s):  
Nilpotent Group ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Free Nilpotent Group

We prove that a subgroup of a finitely generated free nilpotent group F is undistorted if and only if it is a retract of a subgroup of finite index in F.

The Structure of 𝐺-Free Nilpotent Groups of Step 3

2004 ◽  
Vol 11 (1) ◽  
pp. 27-33
Author(s):  
M. Amaglobeli
Keyword(s):  
Nilpotent Group ◽  
Canonical Form ◽  
Nilpotent Groups ◽  
Free Nilpotent Group

Abstract The canonical form of elements of a 𝐺-free nilpotent group of step 3 is defined assuming that the group 𝐺 contains no elements of order 2.

scholarly journals The binomial equivalence classes of finite words

2020 ◽  
Vol 30 (07) ◽  
pp. 1375-1397
Author(s):  
Marie Lejeune ◽  
Michel Rigo ◽  
Matthieu Rosenfeld
Keyword(s):  
Nilpotent Group ◽  
Equivalence Class ◽  
Free Monoid ◽  
Equivalence Classes ◽  
Single Element ◽  
Free Nilpotent Group ◽  
Context Free ◽  
Abelian Equivalence

Two finite words [Formula: see text] and [Formula: see text] are [Formula: see text]-binomially equivalent if, for each word [Formula: see text] of length at most [Formula: see text], [Formula: see text] appears the same number of times as a subsequence (i.e., as a scattered subword) of both [Formula: see text] and [Formula: see text]. This notion generalizes abelian equivalence. In this paper, we study the equivalence classes induced by the [Formula: see text]-binomial equivalence. We provide an algorithm generating the [Formula: see text]-binomial equivalence class of a word. For [Formula: see text] and alphabet of [Formula: see text] or more symbols, the language made of lexicographically least elements of every [Formula: see text]-binomial equivalence class and the language of singletons, i.e., the words whose [Formula: see text]-binomial equivalence class is restricted to a single element, are shown to be non-context-free. As a consequence of our discussions, we also prove that the submonoid generated by the generators of the free nil-[Formula: see text] group (also called free nilpotent group of class [Formula: see text]) on [Formula: see text] generators is isomorphic to the quotient of the free monoid [Formula: see text] by the [Formula: see text]-binomial equivalence.

The Group Ring Of a Class Of Infinite Nilpotent Groups

1955 ◽  
Vol 7 ◽  
pp. 169-187 ◽  
Author(s):  
S. A. Jennings
Keyword(s):  
Nilpotent Group ◽  
Group Ring ◽  
Characteristic Zero ◽  
Discrete Group ◽  
Nilpotent Groups ◽  
Zero Element ◽  
Free Nilpotent Group ◽  
Image Position ◽  
Torsion Free ◽  
The Ideal

Introduction. In this paper we study the (discrete) group ring Γ of a finitely generated torsion free nilpotent group over a field of characteristic zero. We show that if Δ is the ideal of Γ spanned by all elements of the form G − 1, where G ∈ , thenand the only element belonging to Δw for all w is the zero element (cf. (4.3) below).

scholarly journals On nilpotent and polycyclic groups

1989 ◽  
Vol 40 (1) ◽  
pp. 119-122
Author(s):  
Robert J. Hursey
Keyword(s):  
Nilpotent Group ◽  
Central Series ◽  
Finitely Generated ◽  
Free Nilpotent Group ◽  
Polycyclic Groups ◽  
Torsion Free

A group G is torsion-free, finitely generated, and nilpotent if and only if G is a supersolvable R-group. An ordered polycylic group G is nilpotent if and only if there exists an order on G with respect to which the number of convex subgroups is one more than the length of G. If the factors of the upper central series of a torsion-free nilpotent group G are locally cyclic, then consecutive terms of the series are jumps, and the terms are absolutely convex subgroups.

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