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https://doi.org/10.3336/gm.58.2.08

A family of \(2\)-groups and an associated family of semisymmetric, locally \(2\)-arc-transitive graphs

Daniel R Hawtin orcid.org/0000-0002-6466-4282 ; Faculty of Mathematics, University of Rijeka, 51000 Rijeka, Croatia
Cheryl E Praeger orcid.org/0000-0002-0881-7336 ; Department of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia
Jin-Xin Zhou ; School of Mathematics and Statistics, Beijing Jiaotong University, Beijing 100044, P.R. China

Preuzmi JATS datoteku

Sažetak

A mixed dihedral group is a group \(H\) with two disjoint subgroups \(X\) and \(Y\), each elementary abelian of order \(2^n\), such that \(H\) is generated by \(X\cup Y\), and \(H/H'\cong X\times Y\). In this paper, for each \(n\geq 2\), we construct a mixed dihedral \(2\)-group \(H\) of nilpotency class \(3\) and order \(2^a\) where \(a=(n^3+n^2+4n)/2\), and a corresponding graph \(\Sigma\), which is the clique graph of a Cayley graph of \(H\). We prove that \(\Sigma\) is semisymmetric, that is, \({\mathop{\rm Aut}}(\Sigma)\) acts transitively on the edges but intransitively on the vertices of \(\Sigma\). These graphs are the first known semisymmetric graphs constructed from groups that are not \(2\)-generated (indeed \(H\) requires \(2n\) generators). Additionally, we prove that \(\Sigma\) is locally \(2\)-arc-transitive, and is a normal cover of the `basic' locally \(2\)-arc-transitive graph \({\rm\bf K}_{2^n,2^n}\). As such, the construction of this family of graphs contributes to the investigation of normal covers of prime-power order of basic locally \(2\)-arc-transitive graphs – the `local' analogue of a question posed by C. H. Li.

Ključne riječi

Semisymmetric, \(2\)-arc-transitive, edge-transitive, normal cover, Cayley graph

Hrčak ID:

312015

URI

https://hrcak.srce.hr/312015

Datum izdavanja:

3.1.2025.

Posjeta: 500 *