scholarly journals NORMAL AUTOMORPHISMS OF A FREE METABELIAN NILPOTENT GROUP

2009 ◽  
Vol 52 (1) ◽  
pp. 169-177
Author(s):  
GÉRARD ENDIMIONI
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Soluble Group ◽  
Groups Of Automorphisms ◽  
Inner Automorphisms ◽  
Free Soluble Group

AbstractAn automorphism φ of a group G is said to be normal if φ(H) = H for each normal subgroup H of G. These automorphisms form a group containing the group of inner automorphisms. When G is a non-abelian free (or free soluble) group, it is known that these groups of automorphisms coincide, but this is not always true when G is a free metabelian nilpotent group. The aim of this paper is to determine the group of normal automorphisms in this last case.

scholarly journals THE GROUPS OF AUTOMORPHISMS ARE COMPLETE FOR FREE BURNSIDE GROUPS OF ODD EXPONENTS n ≥ 1003

2013 ◽  
Vol 23 (06) ◽  
pp. 1485-1496 ◽  
Author(s):  
V. S. ATABEKYAN
Keyword(s):  
Normal Subgroup ◽  
Free Groups ◽  
Group Of Automorphisms ◽  
Burnside Groups ◽  
Free Burnside Group ◽  
Trivial Center ◽  
Groups Of Automorphisms ◽  
Group B ◽  
Inner Automorphisms ◽  
Automorphism Tower

It is proved that the group of automorphisms Aut (B(m, n)) of the free Burnside group B(m, n) is complete for every odd exponent n ≥ 1003 and for any m > 1, that is, it has a trivial center and any automorphism of Aut (B(m, n)) is inner. Thus, the automorphism tower problem for groups B(m, n) is solved and it is showed that it is as short as the automorphism tower of the absolutely free groups. Moreover, the group of all inner automorphisms Inn (B(m, n)) is the unique normal subgroup in Aut (B(m, n)) among all its subgroups, which are isomorphic to free Burnside group B(s, n) of some rank s.

scholarly journals Cyclically separated groups

1982 ◽  
Vol 26 (3) ◽  
pp. 355-384 ◽  
Author(s):  
Brian Hartley ◽  
John C. Lennox ◽  
Akbar H. Rhemtulla
Keyword(s):  
Normal Subgroup ◽  
Finite Index ◽  
Finite Rank ◽  
Soluble Group ◽  
Residually Finite Groups ◽  
Interesting Class ◽  
Isolated Subgroup ◽  
Free Section ◽  
Torsion Free ◽  
Free Soluble Group

We call a group G cyclically separated if for any given cyclic subgroup B in G and subgroup A of finite index in B, there exists a normal subgroup N of G of finite index such that N ∩ B = A. This is equivalent to saying that for each element x ∈ G and integer n ≥ 1 dividing the order o(x) of x, there exists a normal subgroup N of G of finite index such that Nx has order n in G/N. As usual, if x has infinite order then all integers n ≥ 1 are considered to divide o(x). Cyclically separated groups, which are termed “potent groups” by some authors, form a natural subclass of residually finite groups and finite cyclically separated groups also form an interesting class whose structure we are able to describe reasonably well. Construction of finite soluble cyclically separated groups is given explicitly. In the discussion of infinite soluble cyclically separated groups we meet the interesting class of Fitting isolated groups, which is considered in some detail. A soluble group G of finite rank is Fitting isolated if, whenever H = K/L (L ⊲ K ≤ G) is a torsion-free section of G and F(H) is the Fitting subgroup of H then H/F(H) is torsion-free abelian. Every torsion-free soluble group of finite rank contains a Fitting isolated subgroup of finite index.

scholarly journals Lower central depth in finitely generated soluble-by-finite groups

1978 ◽  
Vol 19 (2) ◽  
pp. 153-154 ◽  
Author(s):  
John C. Lennox
Keyword(s):  
Finite Number ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Depth ◽  
Lower Central Series ◽  
Central Series ◽  
Finitely Generated ◽  
Central Depth

We say that a group G has finite lower central depth (or simply, finite depth) if the lower central series of G stabilises after a finite number of steps.In [1], we proved that if G is a finitely generated soluble group in which each two generator subgroup has finite depth then G is a finite-by-nilpotent group. Here, in answer to a question of R. Baer, we prove the following stronger version of this result.

A Class of Three-Generator, Three-Relation, Finite Groups

1970 ◽  
Vol 22 (1) ◽  
pp. 36-40 ◽  
Author(s):  
J. W. Wamsley
Keyword(s):  
Nilpotent Group ◽  
Finite Groups ◽  
Soluble Group ◽  
Finite Soluble Group ◽  
Finite Nilpotent Group ◽  
Image Position

Mennicke (2) has given a class of three-generator, three-relation finite groups. In this paper we present a further class of three-generator, threerelation groups which we show are finite.The groups presented are defined as:with α|γ| ≠ 1, β|γ| ≠ 1, γ ≠ 0.We prove the following result.THEOREM 1. Each of the groups presented is a finite soluble group.We state the following theorem proved by Macdonald (1).THEOREM 2. G1(α, β, 1) is a finite nilpotent group.1. In this section we make some elementary remarks.

scholarly journals On groups with all subgroups almost subnormal

2004 ◽  
Vol 77 (2) ◽  
pp. 165-174 ◽  
Author(s):  
Eloisa Detomi
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Finite Index ◽  
Normal Closure ◽  
Fixed Integer ◽  
Torsion Free

AbstractIn this paper we consider groups in which every subgroup has finite index in the nth term of its normal closure series, for a fixed integer n. We prove that such a group is the extension of a finite normal subgroup by a nilpotent group, whose class is bounded in terms of n only, provided it is either periodic or torsion-free.

On the automorphism group of the integral group ring of the infinite dihedral group

1988 ◽  
Vol 108 (1-2) ◽  
pp. 1-10
Author(s):  
D. A. R. Wallace
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Group Ring ◽  
Polynomial Ring ◽  
Dihedral Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Index 2 ◽  
Skolem Theorem ◽  
Inner Automorphisms

SynopsisLet ℤ(G) be the integral group ring of the infinite dihedral groupG; the aim is to calculate Autℤ(G), the group of Z-linear automorphisms of ℤ(G). It is shown that Aut ℤ(G), the subgroup of Aut *ℤ(G) consisting of those automorphisms that preserve elementwise the centre of ℤ(G), is a normal subgroup of Aut *ℤ(G) of index 2 and that Aut*ℤ(G) may be embedded monomorphically intoM2(ℤ[t]), the ring of 2 × 2 matrices over a polynomial ring ℤ[t]From this embedding and by the Noether—Skolem Theorem it is shown that Inn (G), the group of inner automorphisms of ℤ(G) induced by the units of ℤ(G), is a normal subgroup of Aut*ℤ(G)/ such that Aut*ℤ(G)/Inn (G) is isomorphic to the Klein four-group.

scholarly journals Fitting quotients of finitely presented abelian-by-nilpotent groups

2014 ◽  
Vol 17 (1) ◽  
pp. 1-12
Author(s):  
J. R. J. Groves ◽  
Ralph Strebel
Keyword(s):  
Normal Subgroup ◽  
Nilpotent Group ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Finitely Presented

Abstract.We show that every finitely generated nilpotent group of class 2 occurs as the quotient of a finitely presented abelian-by-nilpotent group by its largest nilpotent normal subgroup.

ON AUTOMORPHISMS OF THE ENDOMORPHISM SEMIGROUP OF A FREE UNIVERSAL ALGEBRA

2007 ◽  
Vol 17 (05n06) ◽  
pp. 1085-1106 ◽  
Author(s):  
G. MASHEVITZKY ◽  
B. I. PLOTKIN
Keyword(s):  
Lie Algebras ◽  
Universal Algebra ◽  
Automorphism Groups ◽  
Endomorphism Semigroup ◽  
Universal Algebras ◽  
Free Lie Algebras ◽  
Inner Automorphism ◽  
Groups Of Automorphisms ◽  
Inner Automorphisms

Let U be a universal algebra. An automorphism α of the endomorphism semigroup of U defined by α(φ) = sφs-1 for a bijection s : U → U is called a quasi-inner automorphism. We characterize bijections on U defining such automorphisms. For this purpose, we introduce the notion of a pre-automorphism of U. In the case when U is a free universal algebra, the pre-automorphisms are precisely the well-known weak automorphisms of U. We also provide different characterizations of quasi-inner automorphisms of endomorphism semigroups of free universal algebras and reveal their structure. We apply obtained results for describing the structure of groups of automorphisms of categories of free universal algebras, isomorphisms between semigroups of endomorphisms of free universal algebras, automorphism groups of endomorphism semigroups of free Lie algebras etc.

Some Engel conditions on infinite subsets of certain groups

2000 ◽  
Vol 62 (1) ◽  
pp. 141-148 ◽  
Author(s):  
Alireza Abdollahi
Keyword(s):  
Positive Integer ◽  
Nilpotent Group ◽  
Soluble Group ◽  
Metabelian Group ◽  
Infinite Subset ◽  
Central Series ◽  
Finitely Generated ◽  
Locally Nilpotent ◽  
Finite Quotient

Let k be a positive integer. We denote by ɛk(∞) the class of all groups in which every infinite subset contains two distinct elements x, y such that [x,k y] = 1. We say that a group G is an -group provided that whenever X, Y are infinite subsets of G, there exists x ∈ X, y ∈ Y such that [x,k y] = 1. Here we prove that:(1) If G is a finitely generated soluble group, then G ∈ ɛ3(∞) if and only if G is finite by a nilpotent group in which every two generator subgroup is nilpotent of class at most 3.(2) If G is a finitely generated metabelian group, then G ∈ ɛk(∞) if and only if G/Zk (G) is finite, where Zk (G) is the (k + 1)-th term of the upper central series of G.(3) If G is a finitely generated soluble ɛk(∞)-group, then there exists a positive integer t depending only on k such that G/Zt (G) is finite.(4) If G is an infinite -group in which every non-trivial finitely generated subgroup has a non-trivial finite quotient, then G is k-Engel. In particular, G is locally nilpotent.

ON THE AUTOMORPHISM GROUP OF A FREE METABELIAN LIE ALGEBRA

2008 ◽  
Vol 18 (02) ◽  
pp. 209-226 ◽  
Author(s):  
VITALY ROMAN'KOV
Keyword(s):  
Automorphism Group ◽  
Normal Subgroup ◽  
Lie Algebra ◽  
Free Subgroup ◽  
Finite Set ◽  
Free Metabelian Lie Algebra ◽  
Inner Automorphisms ◽  
Infinite Set ◽  
Wild Automorphisms

Let K be a field of any characteristic. We prove that a free metabelian Lie algebra M3 of rank 3 over K admits wild automorphisms. Moreover, the subgroup I Aut M3 of all automorphisms identical modulo the derived subalgebra [Formula: see text] cannot be generated by any finite set of IA-automorphisms together with the sets of all inner and all tame IA-automorphisms. In the case if K is finite the group Aut M3 cannot be generated by any finite set of automorphisms together with the sets of all tame, all inner automorphisms and all one-row automorphisms. We present an infinite set of wild IA-automorphisms of M3 which generates a free subgroup F∞ modulo normal subgroup generated by all tame, all inner and all one-row automorphisms of M3.

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