In particular we prove that there is no solvable group whose bipartite divisor graph is $C_{4}+C_{6}$.
📌 3 Key Takeaways
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Objective & Hypothesis:Given a finite group $G$, the \textit{bipartite divisor graph}, denoted by $B(G)$, for its irreducible character degrees is the bipartite graph with bipartition consisting of $cd(G)^{*}$, where $cd(G)^{*}$ denotes the nonidentity irreducible character degrees of $G$ and the $\rho(G)$ which is the set of prime numbers that divide these degrees, and with $\{p,n\}$ being an edge if $\gcd(p,n)\neq 1$.
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Key Findings & Data:Group Theory, 2017], the author considered the cases where $B(G)$ is a path or a cycle and discussed some properties of $G$.
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Conclusion & Impact:In particular we prove that there is no solvable group whose bipartite divisor graph is $C_{4}+C_{6}$.
🔬Design:Research Article
📍Scope:2017 Verileri
⚙️Variables:bipartite divisor graph · irreducible character degrees · regular graph
Abstract
Given a finite group $G$, the \textit{bipartite divisor graph}, denoted by $B(G)$, for its irreducible character degrees is the bipartite graph with bipartition consisting of $cd(G)^{*}$, where $cd(G)^{*}$ denotes the nonidentity irreducible character degrees of $G$ and the $\rho(G)$ which is the set of prime numbers that divide these degrees, and with $\{p,n\}$ being an edge if $\gcd(p,n)\neq 1$. In [Bipartite divisor graph for the set of irreducible character degress, Int. J. Group Theory, 2017], the author considered the cases where $B(G)$ is a path or a cycle and discussed some properties of $G$. In particular she proved that $B(G)$ is a cycle if and only if $G$ is solvable and $B(G)$ is either a cycle of length four or six. Inspired by $2$-regularity of cycles, in this paper we consider the case where $B(G)$ is an $n$-regular graph for $n\in\{1,2,3\}$. In particular we prove that there is no solvable group whose bipartite divisor graph is $C_{4}+C_{6}$.
Hafezieh, R. (2019). On regular bipartite divisor graph for the set of irreducible character degrees. Hacettepe Journal of Mathematics and Statistics, Volume: 48 Issue: 6. https://ajindex.com/articles/1402705
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