Properties of the Secondary Hochschild Homology

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.Comment: 15 page

Classifying Families of Character Degree Graphs of Solvable Groups

We investigate prime character degree graphs of solvable groups. In particular, we consider a family of graphs $\Gamma_{k,t}$ constructed by adjoining edges between two complete graphs in a one-to-one fashion. In this paper we determine completely which graphs $\Gamma_{k,t}$ occur as the prime character degree graph of a solvable group.Comment: 7 pages, 5 figures, updated version is reorganize

On the Absence of a Normal Nonabelian Sylow Subgroup

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.Comment: 6 pages, 1 figur

Classifying character degree graphs with seven vertices

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.Comment: 28 pages, 18 figures. arXiv admin note: text overlap with arXiv:2108.0833

A Simplicial Construction for Noncommutative Settings

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.Comment: 9 pages, all comments are welcom