scholarly journals Automata groups generated by Cayley machines of groups of nilpotency class two

2021 ◽  
pp. 1-13
Author(s):  
Ning Yang
Keyword(s):  
Finite Groups ◽  
Nilpotency Class ◽  
Automata Groups ◽  
Lamplighter Groups

We build presentations for automata groups generated by Cayley machines of finite groups of nilpotency class two and prove that these automata groups are all cross-wired lamplighter groups.

scholarly journals Group Laws Implying Virtual Nilpotence

2003 ◽  
Vol 74 (3) ◽  
pp. 295-312 ◽  
Author(s):  
R. G. Burns ◽  
Yuri Medvedev
Keyword(s):  
Large Class ◽  
Finite Groups ◽  
Additional Condition ◽  
Metabelian Group ◽  
Nilpotency Class ◽  
Finitely Generated ◽  
Finitely Generated Groups ◽  
Locally Graded Groups ◽  
Finite Exponent ◽  
Group Law

AbstractIf ω ≡ 1 is a group law implying virtual nilpotence in every finitely generated metabelian group satisfying it, then it implies virtual nilpotence for the finitely generated groups of a large class of groups including all residually or locally soluble-or-finite groups. In fact the groups of satisfying such a law are all nilpotent-by-finite exponent where the nilpotency class and exponent in question are both bounded above in terms of the length of ω alone. This yields a dichotomy for words. Finally, if the law ω ≡ 1 satisfies a certain additional condition—obtaining in particular for any monoidal or Engel law—then the conclusion extends to the much larger class consisting of all ‘locally graded’ groups.

scholarly journals Word problems for finite nilpotent groups

2020 ◽  
Vol 115 (6) ◽  
pp. 599-609
Author(s):  
Rachel D. Camina ◽  
Ainhoa Iñiguez ◽  
Anitha Thillaisundaram
Keyword(s):  
Finite Group ◽  
Nilpotent Group ◽  
Finite Groups ◽  
Irreducible Character ◽  
Nilpotent Groups ◽  
Nilpotency Class ◽  
Character Degrees ◽  
Character Theory ◽  
Odd Order ◽  
Related Group

AbstractLet w be a word in k variables. For a finite nilpotent group G, a conjecture of Amit states that $$N_w(1)\ge |G|^{k-1}$$ N w ( 1 ) ≥ | G | k - 1 , where for $$g\in G$$ g ∈ G , the quantity $$N_w(g)$$ N w ( g ) is the number of k-tuples $$(g_1,\ldots ,g_k)\in G^{(k)}$$ ( g 1 , … , g k ) ∈ G ( k ) such that $$w(g_1,\ldots ,g_k)={g}$$ w ( g 1 , … , g k ) = g . Currently, this conjecture is known to be true for groups of nilpotency class 2. Here we consider a generalized version of Amit’s conjecture, which states that $$N_w(g)\ge |G|^{k-1}$$ N w ( g ) ≥ | G | k - 1 for g a w-value in G, and prove that $$N_w(g)\ge |G|^{k-2}$$ N w ( g ) ≥ | G | k - 2 for finite groups G of odd order and nilpotency class 2. If w is a word in two variables, we further show that the generalized Amit conjecture holds for finite groups G of nilpotency class 2. In addition, we use character theory techniques to confirm the generalized Amit conjecture for finite p-groups (p a prime) with two distinct irreducible character degrees and a particular family of words. Finally, we discuss the related group properties of being rational and chiral, and show that every finite group of nilpotency class 2 is rational.

scholarly journals Identical relations and decision procedures for groups

1967 ◽  
Vol 7 (1) ◽  
pp. 39-47 ◽  
Author(s):  
Hermann Heineken ◽  
Peter M. Neumann
Keyword(s):  
Structural Property ◽  
Finite Groups ◽  
Decision Procedures ◽  
Nilpotency Class ◽  
Cartesian Products ◽  
Derived Length

Although varieties of groups can in theory be determined as well by the identical relations which the groups all satisfy as by some structural property inherited by subgroups, factor groups and cartesian products which the groups have in common, it seems in practice just as hard to answer questions about properties of a group from knowledge of identical relations as it is from, say, a presentation. Many of the important questions connected with Burnside's problems exemplify this difficulty: we still do not know if there is a bound on the derived length of finite groups of exponent 4, nor whether there is a bound on the nilpotency class of finite groups of exponent p (p ≧ 5, a fixed prime).

Another Existence and Uniqueness Proof of the Tits Group

2008 ◽  
Vol 15 (02) ◽  
pp. 241-278
Author(s):  
Gerhard O. Michler ◽  
Lizhong Wang
Keyword(s):  
Finite Groups ◽  
Existence And Uniqueness ◽  
Frobenius Group ◽  
Nilpotency Class ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Uniqueness Proof ◽  
Groups Of Lie Type ◽  
Uniqueness Criteria ◽  
Tits Group

In this article we give a self-contained existence and uniqueness proof for the Tits simple group T. Parrott gave the first uniqueness proof. Whereas Tits' and Parrott's results employ the theory of finite groups of Lie type, our existence and uniqueness proof follows from the general algorithms and uniqueness criteria for abstract finite simple groups described in the first author's book [11]. All we need from the previous papers is the fact that the centralizer H of the Tits group T is an extension of a 2-group J with order 29 and nilpotency class 3 by a Frobenius group F of order 20 such that the center Z(H) has order 2 and any Sylow 5-subgroup Q of H has a centralizer CJ(Q) ≤ Z(H).

scholarly journals K -homology and K -theory for the lamplighter groups of finite groups

2017 ◽  
Vol 115 (6) ◽  
pp. 1207-1226 ◽  
Author(s):  
Ramón Flores ◽  
Sanaz Pooya ◽  
Alain Valette
Keyword(s):  
Finite Groups ◽  
Lamplighter Groups ◽  
K Theory

scholarly journals The degree of commutativity and lamplighter groups

2018 ◽  
Vol 28 (07) ◽  
pp. 1163-1173 ◽  
Author(s):  
Charles Garnet Cox
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Wreath Products ◽  
Infinite Groups ◽  
Lamplighter Groups

The degree of commutativity of a group [Formula: see text] measures the probability of choosing two elements in [Formula: see text] which commute. There are many results studying this for finite groups. In [Y. Antolín, A. Martino and E. Ventura, Degree of commutativity of infinite groups, Proc. Amer. Math. Soc. 145 (2017) 479–485, MR 3577854], this was generalized to infinite groups. In this note, we compute the degree of commutativity for wreath products of the form [Formula: see text] and [Formula: see text], where [Formula: see text] is any finite group.

scholarly journals Intense Automorphisms of Finite Groups

2021 ◽  
Vol 273 (1341) ◽  
Author(s):  
Mima Stanojkovski
Keyword(s):  
Prime Number ◽  
Semidirect Product ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
Nilpotency Class ◽  
Content Type ◽  
Formula Group ◽  
Special Case

Let G G be a group. An automorphism of G G is called intense if it sends each subgroup of G G to a conjugate; the collection of such automorphisms is denoted by Int ⁡ ( G ) \operatorname {Int}(G) . In the special case in which p p is a prime number and G G is a finite p p -group, one can show that Int ⁡ ( G ) \operatorname {Int}(G) is the semidirect product of a normal p p -Sylow and a cyclic subgroup of order dividing p − 1 p-1 . In this paper we classify the finite p p -groups whose groups of intense automorphisms are not themselves p p -groups. It emerges from our investigation that the structure of such groups is almost completely determined by their nilpotency class: for p > 3 p>3 , they share a quotient, growing with their class, with a uniquely determined infinite 2-generated pro- p p group.

scholarly journals Finite coverings by subgroups with a given property

1993 ◽  
Vol 35 (2) ◽  
pp. 179-188 ◽  
Author(s):  
Marc A. Brodie ◽  
Luise-Charlotte Kappe
Keyword(s):  
Finite Index ◽  
Finite Groups ◽  
Nilpotency Class ◽  
Finite Covering ◽  
Theoretic Property ◽  
Abelian Subgroups

Let be a group-theoretic property. We say a group has a finite covering by -subgroups if it is the set-theoretic union of finitely many -subgroups. The topic of this paper is the investigation of groups having a finite covering by nilpotent subgroups, n-abelian subgroups or 2-central subgroups.R. Baer [12; 4.16] characterized central-by-finite groups as those groups having a finite covering by abelian subgroups. In [6] it was shown that [G: ZC (G)] finite implies the existence of a finite covering by subgroups of nilpotency class c, i.e. ℜc-groups. However, an example of a group is given there which has a finite covering by ℜ2-groups, but Z2(G) does not have finite index in the group. These results raise two questions, on which we will focus our investigations.

Sign in / Sign up

Export Citation Format

Share Document