Ordinary G-permutable subgroups of finite simple groups

Abstract

We completely solve part (a) of Problem 17.112 in the Kourovka Notebook. We prove that a non-abelian finite simple group G has a proper non-trivial ordinary G-permutable subgroup if and only if G is isomorphic to PSL₂(q) where q≥4 is a prime power and qnot≡3pmod4, ²B₂(2^2a+1) where a≥1, J₁, Sp₄(4), or PSp₄(p) where p is prime and p≡1,13,17pmod24. Moreover, we classify all proper non-trivial ordinary G-permutable subgroups of these groups. Every other non-abelian finite simple group G has no proper non-trivial ordinary G-permutable subgroup.

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BibTeX RIS

BibTeX

@misc{zhang2026ordinary,
  title = {{Ordinary $G$-permutable subgroups of finite simple groups}},
  author = {Shengmin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12329},
  eprint = {2610.12329},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12329},
  note = {CC0},
}

RIS

TY  - UNPB
TI  - Ordinary G-permutable subgroups of finite simple groups
AU  - Shengmin Zhang
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12329
UR  - https://arxiv.org/abs/2610.12329
AB  - We completely solve part (a) of Problem 17.112 in the Kourovka Notebook. We prove that a non-abelian finite simple group G has a proper non-trivial ordinary G-permutable subgroup if and only if G is isomorphic to PSL_2(q) where q≥4 is a prime power and qnot≡3pmod4, ^2B_2(2^2a+1) where a≥1, J_1, Sp_4(4), or PSp_4(p) where p is prime and p≡1,13,17pmod24. Moreover, we classify all proper non-trivial ordinary G-permutable subgroups of these groups. Every other non-abelian finite simple group G has no proper non-trivial ordinary G-permutable subgroup.
ER  -

Abstract

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