Generators of the automorphism group of a free nilpotent group

1976 ◽  
Vol 15 (4) ◽  
pp. 289-292 ◽  
Author(s):  
A. V. Goryaga
1981 ◽  
Vol 90 (3) ◽  
pp. 403-409
Author(s):  
U. H. M. Webb

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.


1976 ◽  
Vol 79 (2) ◽  
pp. 271-279 ◽  
Author(s):  
Joan L. Dyer ◽  
Edward Formanek

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.


2010 ◽  
Vol 20 (05) ◽  
pp. 661-669 ◽  
Author(s):  
TARA C. DAVIS

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.


2004 ◽  
Vol 11 (1) ◽  
pp. 27-33
Author(s):  
M. Amaglobeli

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.


2020 ◽  
Vol 30 (07) ◽  
pp. 1375-1397
Author(s):  
Marie Lejeune ◽  
Michel Rigo ◽  
Matthieu Rosenfeld

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.


1955 ◽  
Vol 7 ◽  
pp. 169-187 ◽  
Author(s):  
S. A. Jennings

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).


1989 ◽  
Vol 40 (1) ◽  
pp. 119-122
Author(s):  
Robert J. Hursey

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.


Sign in / Sign up

Export Citation Format

Share Document