On homotopy nilpotency of the octonian plane $\mathbb{O}P^2$

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.

On the modular isomorphism problem for groups of class 3 and obelisks

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.

Intense Automorphisms of Finite Groups

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.

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

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.

The upper central series of the maximal normal 𝑝-subgroup of a group of automorphisms

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) .

Diameters of random Cayley graphs of finite nilpotent groups

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.