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A family of graphs that cannot occur as character degree graphs of solvable groups
Published 27/02/2024
arXiv.org
We investigate character degree graphs of solvable groups. In particular, we provide general results that can be used to eliminate which degree graphs can occur as solvable groups. Finally, we show a specific family of graphs cannot occur as a character degree for any solvable group.
Other
Classifying character degree graphs with seven vertices
Published 02/08/2023
arXiv.org
We study here the graphs with seven vertices in an effort to classify which of them appear as the prime character degree graphs of finite solvable groups. This classification is complete for the disconnected graphs. Of the 853 non-isomorphic connected graphs, we were able to demonstrate that twenty-two occur as prime character degree graphs. Two are of diameter three, while the remaining are constructed as direct products. Forty-four graphs remain unclassified.
Other
Secondary Hochschild cohomology and derivations
Published 22/02/2023
arXiv.org
In this paper, we introduce a generalization of derivations. Using these so-called secondary derivations, along with an analogue of Connes' Long Exact Sequence, we are able to provide computations in low dimension for the secondary Hochschild and cyclic cohomologies associated to a commutative triple. We then establish a universal property, which paves the way to relating secondary K\"ahler differentials with the aforementioned secondary derivations.
Other
Secondary Hochschild homology and differentials
Published 23/06/2022
arXiv.org
In this paper we study a generalization of K\"ahler differentials, which correspond to the secondary Hochschild homology associated to a triple \((A,B,\varepsilon)\). We establish computations in low dimension, while also showing how this connects with the kernel of a multiplication map.
Other
On prime character degree graphs occurring within a family of graphs (ii)
Published 18/08/2021
arXiv.org
In this paper, we continue the classification work done in the first paper of the same name. With careful modifications of our previous approach, we are able to deduce (with two notable exceptions) which members of the previously introduced graph family manifest as the prime character degree graph of some solvable group.
Other
A Simplicial Construction for Noncommutative Settings
Published 30/05/2019
arXiv.org
In this paper we present a general construction that can be used to define the higher Hochschild homology for a noncommutative algebra. We also discuss other examples where this construction can be used.
Other
On the Absence of a Normal Nonabelian Sylow Subgroup
Published 18/03/2018
arXiv.org
Let \(G\) be a finite solvable group. We show that \(G\) does not have a normal nonabelian Sylow \(p\)-subgroup when its prime character degree graph \(\Delta(G)\) satisfies a technical hypothesis.
Other
Properties of the Secondary Hochschild Homology
Published 07/05/2017
arXiv.org
In this paper we study properties of the secondary Hochschild homology of the triple \((A,B,\varepsilon)\) with coefficients in \(M\). We establish a type of Morita equivalence between two triples and show that \(H_\bullet((A,B,\varepsilon);M)\) is invariant under this equivalence. We also prove the existence of an exact sequence which connects the usual and the secondary Hochschild homologies in low dimension, allowing one to perform easy computations. The functoriality of \(H_\bullet((A,B,\varepsilon);M)\) is also discussed.
Other
Bar Simplicial Modules and Secondary Cyclic (Co)homology
Published 11/05/2016
arXiv.org
In this paper we study the simplicial structure of the complex \(C^{\bullet}((A,B,\varepsilon); M)\), associated to the secondary Hochschild cohomology. The main ingredient is the simplicial object \(\mathcal{B}(A,B,\varepsilon)\), which plays a role equivalent to that of the bar resolution associated to an algebra. We also introduce the secondary cyclic (co)homology and establish some of its properties (Theorems 3.9 and 4.11).