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Title: homogeneous spaces of semidirect products and finite gelfand pairs.
Abstract: Let $K\leq H$ be two finite groups and let $C\leq A$ be two finite abelian groups, with $H$ acting on $A$ as a group of isomorphisms admitting $C$ as a $K$-invariant subgroup. We study the homogeneous space $X\coloneqq\left(H\ltimes A\right)/\left(K\ltimes C\right)$ and determine the decomposition of the permutation representation of $H\ltimes A$ acting on $X$. We then characterize when this is multiplicity-free, that is, when $\left(H\ltimes A,K\ltimes C\right)$ is a Gelfand pair. If this is the case, we explicitly calculate the corresponding spherical functions. From our general construction and related analysis, we recover Dunkl's results on the $q$-analog of the nonbinary Johnson scheme.
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In conclusion, researchers introduced a mathematical explanation for the rise of Fourier features in learning systems like neural networks. Also, they proved that if a machine learning model of a specific kind is invariant to a finite group, then its weights are closely related to the Fourier transform on that group, and the algebraic structure of an unknown group can be recovered from an invariant model. Future work includes the study of analogs of the proposed theory on real numbers which is an interesting area that will be aligned more towards the current practices in the field.
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Forbidden subgraphs in enhanced power graphs of finite groups
- Original Paper
- Published: 14 May 2024
- Volume 118 , article number 110 , ( 2024 )
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- Xuanlong Ma 1 ,
- Samir Zahirović ORCID: orcid.org/0000-0001-7719-3996 2 ,
- Yubo Lv 3 &
- Yanhong She 1
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The enhanced power graph of a group is the simple graph whose vertex set is consisted of all elements of the group, and whose any pair of vertices are adjacent if they generate a cyclic subgroup. In this paper, we classify all finite groups whose enhanced power graphs are split and threshold. We also classify all finite nilpotent groups whose enhanced power graphs are chordal graphs and cographs. Finally, we give some families of non-nilpotent groups whose enhanced power graphs are chordal graphs and cographs. These results partly answer a question posed by Peter J. Cameron.
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Certain properties of the enhanced power graph associated with a finite group
On co-maximal subgroup graph of a group-ii, finite groups whose character degree graphs coincide with their prime graphs, data availability.
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Xuanlong Ma & Yanhong She
Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad, Novi Sad, 21000, Serbia
Samir Zahirović
School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550001, China
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Xuanlong Ma’s research is supported by National Natural Science Foundation of China (Grant No. 12326333) and Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 22JSQ024). Samir Zahirović acknowledges financial support of the Ministry of Education, Science and Technological Development of the Republic of Serbia (Grant No. 451-03-68/2022-14/200125). Yanhong She’s research is supported by National Natural Science Foundation of China (Grant No. 61976244).
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Ma, X., Zahirović, S., Lv, Y. et al. Forbidden subgraphs in enhanced power graphs of finite groups. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118 , 110 (2024). https://doi.org/10.1007/s13398-024-01611-1
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DOI : https://doi.org/10.1007/s13398-024-01611-1
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