A proof of the Galt--Tyutyanov conjecture on Skiba's question

@misc{zhang2026proof,
  title = {{A proof of the Galt--Tyutyanov conjecture on Skiba's question}},
  author = {Shengmin Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12295},
  eprint = {2610.12295},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12295},
  note = {CC0},
}

TY  - UNPB
TI  - A proof of the Galt--Tyutyanov conjecture on Skiba's question
AU  - Shengmin Zhang
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12295
UR  - https://arxiv.org/abs/2610.12295
AB  - A subgroup A of a finite group G is \emph{hereditarily G-permutable} if, for every subgroup E with A≤ E≤ G and every subgroup B of E, some G-conjugate of B permutes with A in E. We give a new criterion for a finite group to be simple. A nontrivial finite group G is simple if and only if it has no proper nontrivial hereditarily G-permutable subgroup. This proves the conjecture of Galt and Tyutyanov and answers Skiba's question in part \textup{(b)} of Problem 17.112 in the Kourovka Notebook. Earlier work settled the conjecture for alternating groups, sporadic groups, exceptional groups of Lie type, and the groups PSL_2(q) and PSU_3(q). We prove the conjecture for all classical groups. The proof uses descent through parabolic subgroups and the classification of factorizations of finite simple groups.
ER  -