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Simplicial complexes associated to character degrees of solvable groups

Authors

Aleksandra Kostić, Nela Milošević, Zoran Z. Petrović
Type: Article
Publication Date: 2025-05-05
Citations: 0
DOI: https://doi.org/10.1142/s0219498826502312

Abstract

Graphs associated to the set of irreducible character degrees of a finite group [Formula: see text] have been extensively studied as a way of understanding structure of the underlying group. Another approach, proposed by Isaacs, is to study associated simplicial complexes, namely, the common divisor simplicial complex [Formula: see text] and the prime divisor simplicial complex [Formula: see text]. These complexes can be associated to any set of positive integers and this paper shows that they are homotopy equivalent. Further, considering these complexes associated to the set of irreducible character degrees, we give a bound on the rank of the fundamental group.

Locations

  • Journal of Algebra and Its Applications
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We define two different simplicial complexes, the common divisor simplicial complex and the prime divisor simplicial complex, from a set of integers, and explore their similarities. We will define a … We define two different simplicial complexes, the common divisor simplicial complex and the prime divisor simplicial complex, from a set of integers, and explore their similarities. We will define a map between the two simplicial complexes, and use this map to show that for any set of integers, the fundamental groups of the resulting simplicial complexes are isomorphic.

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Alexander Grigor’yan , Yu. V. Muranov , Shing‐Tung Yau
The cohomology of digraphs was introduced for the first time by Dimakis and Müller-Hoissen.Their algebraic definition is based on a differential calculus on an algebra of functions on the set … The cohomology of digraphs was introduced for the first time by Dimakis and Müller-Hoissen.Their algebraic definition is based on a differential calculus on an algebra of functions on the set of vertices with relations that follow naturally from the structure of the set of edges.A dual notion of homology of digraphs, based on the notion of path complex, was introduced by the authors, and the first methods for computing the (co)homology groups were developed.The interest in homology on digraphs is motivated by physical applications and relations between algebraic and geometrical properties of quivers.The digraph G B of the partially ordered set B S of simplexes of a simplicial complex S has graph homology that is isomorphic to the simplicial homology of S. In this paper, we introduce the concept of cubical digraphs and describe their homology properties.In particular, we define a cubical subgraph G S of G B , whose homologies are isomorphic to the simplicial homologies of S.
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Communicated by Communicated by
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We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial … We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.
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The Fundamental Group of Balanced Simplicial Complexes and Posets

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We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial … We establish an upper bound on the cardinality of a minimal generating set for the fundamental group of a large family of connected, balanced simplicial complexes and, more generally, simplicial posets.

Groups whose bipartite divisor graph for character degrees has four or fewer vertices

2018-09-25
S. A. Moosavi
Let [Formula: see text] be a finite group and [Formula: see text] be the set of nonlinear irreducible character degrees of [Formula: see text]. Suppose that [Formula: see text] is … Let [Formula: see text] be a finite group and [Formula: see text] be the set of nonlinear irreducible character degrees of [Formula: see text]. Suppose that [Formula: see text] is the set of primes dividing some elements of [Formula: see text]. The bipartite divisor graph for [Formula: see text], [Formula: see text], is a graph whose vertices are the disjoint union of [Formula: see text] and [Formula: see text], and a vertex [Formula: see text] is connected to a vertex [Formula: see text] if and only if [Formula: see text]. In this paper, we consider groups whose graph has four or fewer vertices. We show that all these groups are solvable and determine the structure of these groups. We also provide examples of any possible graph.

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Roghayeh Hafezieh , Pablo Spiga
Let $G$ be a finite group. The bipartite divisor graph for the set of irreducible complex character degrees is the undirected graph with vertex set consisting of the prime numbers … Let $G$ be a finite group. The bipartite divisor graph for the set of irreducible complex character degrees is the undirected graph with vertex set consisting of the prime numbers dividing some character degree and of the non-identity character degrees, where a prime number $p$ is declared to be adjacent to a character degree $m$ if and only if $p$ divides $m$. This graph is bipartite and it encodes two of the most widely studied graphs associated to the character degrees of a finite group: the prime graph and the divisor graph on the set of irreducible character degrees. The scope of this paper is two-fold. We draw some attention to the bipartite divisor graph for the set of irreducible complex character degrees by outlining the main results that have been proved so far. In this process we improve some of these results and we leave some open problems.

An overview on the bipartite divisor graph for the set of irreducible character degrees

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Roghayeh Hafezieh , Pablo Spiga
Let $G$ be a finite group. The bipartite divisor graph for the set of irreducible complex character degrees is the undirected graph with vertex set consisting of the prime numbers … Let $G$ be a finite group. The bipartite divisor graph for the set of irreducible complex character degrees is the undirected graph with vertex set consisting of the prime numbers dividing some character degree and of the non-identity character degrees, where a prime number $p$ is declared to be adjacent to a character degree $m$ if and only if $p$ divides $m$. This graph is bipartite and it encodes two of the most widely studied graphs associated to the character degrees of a finite group: the prime graph and the divisor graph on the set of irreducible character degrees. The scope of this paper is two-fold. We draw some attention to the bipartite divisor graph for the set of irreducible complex character degrees by outlining the main results that have been proved so far. In this process we improve some of these results and we leave some open problems.

On the character degree graph of solvable groups

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Let $G$ be a finite solvable group, and let $\Delta (G)$ denote the prime graph built on the set of degrees of the irreducible complex characters of $G$. A fundamental … Let $G$ be a finite solvable group, and let $\Delta (G)$ denote the prime graph built on the set of degrees of the irreducible complex characters of $G$. A fundamental result by P. P. Pálfy asserts that the complement $\bar {\Delta }(G)$ of the graph $\Delta (G)$ does not contain any cycle of length $3$. In this paper we generalize Pálfy's result, showing that $\bar {\Delta }(G)$ does not contain any cycle of odd length, whence it is a bipartite graph. As an immediate consequence, the set of vertices of $\Delta (G)$ can be covered by two subsets, each inducing a complete subgraph. The latter property yields in turn that if $n$ is the clique number of $\Delta (G)$, then $\Delta (G)$ has at most $2n$ vertices. This confirms a conjecture by Z. Akhlaghi and H. P. Tong-Viet, and provides some evidence for the famous $\rho$-$\sigma$ conjecture by B. Huppert.
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