nilpotency class
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Author(s):  
Ammu Elizabeth Antony ◽  
Patali Komma ◽  
Viji Zachariah Thomas
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2021 ◽  
Vol 127 (3) ◽  
Author(s):  
Marek Golasi´nski

Let $\mathbb{O}P^2_{(p)}$ be the $p$-localization of the Cayley projective plane $\mathbb{O}P^2$ for a prime $p$ or $p=0$. We show that the homotopy nilpotency class $\textrm{nil} \Omega(\mathbb{O}P^2_{(p)})<\infty $ for $p>2$ and $\textrm{nil} \Omega (\mathbb{O}P^2_{(p)})=1$ for $p>5$ or $p=0$. The homotopy nilpotency of remaining Rosenfeld projective planes are discussed as well.


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Leo Margolis ◽  
Mima Stanojkovski

Abstract We study the Modular Isomorphism Problem applying a combination of existing and new techniques. We make use of the small group algebra to give a positive answer for two classes of groups of nilpotency class 3. We also introduce a new approach to derive properties of the lower central series of a finite 𝑝-group from the structure of the associated modular group algebra. Finally, we study the class of so-called 𝑝-obelisks which are highlighted by recent computer-aided investigations of the problem.


2021 ◽  
Vol 273 (1341) ◽  
Author(s):  
Mima Stanojkovski

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.


Author(s):  
Ning Yang

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.


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Maria Alicia Aviño ◽  
Phill Schultz ◽  
Marcos Zyman

Abstract Let 𝐺 be a bounded abelian 𝑝-group, with automorphism group Aut ⁡ ( G ) \operatorname{Aut}(G) . Whenever 𝐺 satisfies certain conditions, we determine the upper central series and nilpotency class of the maximal normal 𝑝-subgroup of Aut ⁡ ( G ) \operatorname{Aut}(G) .


2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Daniel El-Baz ◽  
Carlo Pagano

Abstract We prove the existence of a limiting distribution for the appropriately rescaled diameters of random undirected Cayley graphs of finite nilpotent groups of bounded rank and nilpotency class, thus extending a result of Shapira and Zuck which dealt with the case of abelian groups. The limiting distribution is defined on a space of unimodular lattices, as in the case of random Cayley graphs of abelian groups. Our result, when specialised to a certain family of unitriangular groups, establishes a very recent conjecture of Hermon and Thomas. We derive this as a consequence of a general inequality, showing that the diameter of a Cayley graph of a nilpotent group is governed by the diameter of its abelianisation.


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