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PreprintarXiv licencearXiv

Co-intersection graphs of nonabelian finite simple groups have diameter two

  • Henry Bradford
  • Kamilla Rekvényi
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Abstract

Let G be a finite group. The co-intersection graph ΔG c of G has as its vertices the nontrivial proper subgroups of G, with edges joining those pairs of subgroups which intersect trivially. It is clear that every connected component of ΔG c has diameter at most three. In this Note, we show that when T is a nonabelian finite simple group, ΔT is connected of diameter exactly two. We also make several observations about the extent to which a finite group is determined by its co-intersection graph, and about the class of finite graphs arising as co-intersection graphs.

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Cite

BibTeXRIS
BibTeX
@misc{bradford2026cointersection,
  title = {{Co-intersection graphs of nonabelian finite simple groups have diameter two}},
  author = {Henry Bradford and Kamilla Rekvényi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07948},
  eprint = {2609.07948},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07948},
}
RIS
TY  - UNPB
TI  - Co-intersection graphs of nonabelian finite simple groups have diameter two
AU  - Henry Bradford
AU  - Kamilla Rekvényi
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07948
UR  - https://arxiv.org/abs/2609.07948
AB  - Let G be a finite group. The co-intersection graph Δ_G ^c of G has as its vertices the nontrivial proper subgroups of G, with edges joining those pairs of subgroups which intersect trivially. It is clear that every connected component of Δ_G ^c has diameter at most three. In this Note, we show that when T is a nonabelian finite simple group, Δ_T is connected of diameter exactly two. We also make several observations about the extent to which a finite group is determined by its co-intersection graph, and about the class of finite graphs arising as co-intersection graphs.
ER  -