Finite self-similar p-groups with abelian first level stabilizers

We determine all finite p-groups that admit a faithful, self-similar action on the p-ary rooted tree such that the first level stabilizer is abelian. A group is in this class if and only if it is a split extension of an elementary abelian p-group by a cyclic group of order p. The proof is based on use of virtual endomorphisms. In this context the result says that if G is a finite p-group with abelian subgroup H of index p, then there exists a virtual endomorphism of G with trivial core and domain H if and only if G is a split extension of H and H is an elementary abelian p-group.Comment: one direction of theorem 2 extended to regular p-group

Sobre endomorfismos virtuais de p-grupos finitos

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2020.Neste trabalho, estudamos, através dos resultados presentes nos artigos "Finite self-similar pgroups with abelian first level stabilizers" e "On self-similar p-groups", o seguinte problema: quais p-grupos finitos podem ser fielmente representados por um grupo self-similar de automorfismos da árvore p-ária, ou seja, quando um p-grupo finito é self-similar? Responderemos esta pergunta para o caso dos p-grupos finitos que possuem um subgrupo maximal abeliano e para o caso dos p-grupos de classe maximal. Também mostraremos que existem finitos p-grupos self-similar de um dado posto e, consequentemente, finitos p-grupos selfsimilar de uma dada coclasse. Além disso, determinaremos a melhor cota possível para a ordem de um p-grupo self-similar de classe maximal.CNPQIn this work, we study, using the results presented in the articles "Finite self-similar p-groups with abelian first level stabilizers" and "On self-similar p-groups", the following question: which finite p-groups can be faithfully represented as a self-similar group of automorphisms of the p-ary tree, or, in other words, when is a finite p-group self-similar? We will answer this question for the case of finite p-groups whith an abelian maximal subgroup and for the case of p-groups of maximal class. We will also show that there are only finitely many self-similar p-groups of a given rank and, consequently, finitely many self-similar p-groups of a given coclass. Moreover, we will determine the best possible bound for the order of a self-similar p-group of maximal class

On strongly just infinite profinite branch groups

For profinite branch groups, we first demonstrate the equivalence of the Bergman property, uncountable cofinality, Cayley boundedness, the countable index property, and the condition that every non-trivial normal subgroup is open; compact groups enjoying the last condition are called strongly just infinite. For strongly just infinite profinite branch groups with mild additional assumptions, we verify the invariant automatic continuity property and the locally compact automatic continuity property. Examples are then presented, including the profinite completion of the first Grigorchuk group. As an application, we show that many Burger-Mozes universal simple groups enjoy several automatic continuity properties.Comment: Typos and a minor error correcte

Generalized Cluster States Based on Finite Groups

We define generalized cluster states based on finite group algebras in analogy to the generalization of the toric code to the Kitaev quantum double models. We do this by showing a general correspondence between systems with CSS structure and finite group algebras, and applying this to the cluster states to derive their generalization. We then investigate properties of these states including their PEPS representations, global symmetries, and relationship to the Kitaev quantum double models. We also discuss possible applications of these states.Comment: 23 pages, 4 figure

Generalized Color Codes Supporting Non-Abelian Anyons

We propose a generalization of the color codes based on finite groups $G$. For non-abelian groups, the resulting model supports non-abelian anyonic quasiparticles and topological order. We examine the properties of these models such as their relationship to Kitaev quantum double models, quasiparticle spectrum, and boundary structure.Comment: 17 pages, 8 figures; references added, typos remove