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PreprintCC BY 4.0arXiv

Uniform Intersection Numbers and Subgroup-Based Generating Graphs of Groups

  • Projesh Nath Choudhury
  • Biswaranjan Khanra
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Abstract

The uniform intersection number of a finite group G, denoted by ui(G), is the smallest positive integer k such that the intersection of any k distinct maximal subgroups of G is equal to its Frattini subgroup. We determine the uniform intersection number of dicyclic, dihedral, and quasi-dihedral groups. We then investigate the uniform intersection number through the subgroup-based generating graph 𝒢(G) of a finite group G. We characterize the finite groups G for which both the intersection number and the uniform intersection number are equal to two, and show that this condition is equivalent to 𝒢(G) being complete multipartite. We also classify finite groups in terms of the diameter, girth, and perfectness of 𝒢(G). Finally, we show that 𝒢(ℤn) is Hamiltonian if and only if n=p1rp2r, where r≥ 2 and p1 and p2 are distinct primes.

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Cite

BibTeXRIS
BibTeX
@misc{choudhury2026uniform,
  title = {{Uniform Intersection Numbers and Subgroup-Based Generating Graphs of Groups}},
  author = {Projesh Nath Choudhury and Biswaranjan Khanra},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.22268},
  eprint = {2610.22268},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.22268},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Uniform Intersection Numbers and Subgroup-Based Generating Graphs of Groups
AU  - Projesh Nath Choudhury
AU  - Biswaranjan Khanra
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.22268
UR  - https://arxiv.org/abs/2610.22268
AB  - The uniform intersection number of a finite group G, denoted by ui(G), is the smallest positive integer k such that the intersection of any k distinct maximal subgroups of G is equal to its Frattini subgroup. We determine the uniform intersection number of dicyclic, dihedral, and quasi-dihedral groups. We then investigate the uniform intersection number through the subgroup-based generating graph 𝒢(G) of a finite group G. We characterize the finite groups G for which both the intersection number and the uniform intersection number are equal to two, and show that this condition is equivalent to 𝒢(G) being complete multipartite. We also classify finite groups in terms of the diameter, girth, and perfectness of 𝒢(G). Finally, we show that 𝒢(ℤ_n) is Hamiltonian if and only if n=p_1^rp_2^r, where r≥ 2 and p_1 and p_2 are distinct primes.
ER  -