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The automorphism group of the derangement graph of PGL2(q) acting on the projective line

  • Andriaherimanana Sarobidy Razafimahatratra
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Abstract

Given a finite transitive group G≤ Sym(Ω), the derangement graph ΓG is the graph whose vertex set is G, and two vertices g and h are adjacent if the ratio h−1g is a fixed-point-free permutation. In this paper, we show that the automorphism group of the derangement graph of the transitive permutation group corresponding to the natural action of PGL2(q) on the projective line PG1(q) is Aut(ΓPGL2(q)) = (LPGL2(q)× RPGL2(q)) rtimes (⟨ ψ⟩ × γAut(𝔽q)), where LPGL2(q) is the left-regular representation of PGL2(q), RPGL2(q) is the right-regular representation of PGL2(q), γAut(𝔽q) is the group of conjugation by elements of Aut(𝔽q), and ψ: PGL2(q) → PGL2(q) such that ψ(x) = x−1.

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Cite

BibTeXRIS
BibTeX
@misc{razafimahatratra2026automorphism,
  title = {{The automorphism group of the derangement graph of $\operatornamePGL_2(q)$ acting on the projective line}},
  author = {Andriaherimanana Sarobidy Razafimahatratra},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.32969},
  eprint = {2609.32969},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.32969},
}
RIS
TY  - UNPB
TI  - The automorphism group of the derangement graph of PGL_2(q) acting on the projective line
AU  - Andriaherimanana Sarobidy Razafimahatratra
PY  - 2026
DA  - 2026/09/26
PB  - arXiv
DO  - 10.48550/arxiv.2609.32969
UR  - https://arxiv.org/abs/2609.32969
AB  - Given a finite transitive group G≤ Sym(Ω), the derangement graph Γ_G is the graph whose vertex set is G, and two vertices g and h are adjacent if the ratio h^−1g is a fixed-point-free permutation. In this paper, we show that the automorphism group of the derangement graph of the transitive permutation group corresponding to the natural action of PGL_2(q) on the projective line PG_1(q) is Aut(Γ_PGL_2(q)) = (L_PGL_2(q)× R_PGL_2(q)) rtimes (⟨ ψ⟩ × γ_Aut(𝔽_q)), where L_PGL_2(q) is the left-regular representation of PGL_2(q), R_PGL_2(q) is the right-regular representation of PGL_2(q), γ_Aut(𝔽_q) is the group of conjugation by elements of Aut(𝔽_q), and ψ: PGL_2(q) → PGL_2(q) such that ψ(x) = x^−1.
ER  -