PreprintCC BY 4.0arXiv
Sylow synchronization in finite groups: the good case
Abstract
Let G be a finite solvable group such that, for any prime p and any quotient Q of G, there are two Sylow p-subgroups of Q intersecting in Op(Q). Then, for every family (Pi)i=1n of Sylow subgroups of G for distinct primes p1,…,pn, there exists x ∈ G such that Pi ∩ Pix = Opi(G) for all i. This covers groups of odd order, partially settling a conjecture of the second and fourth authors and unifying old results of Bialostocki and Mann on the intersection of nilpotent subgroups. We also prove a result for all symmetric and alternating groups, completing the proof of the conjecture for simple groups as initiated by Burness and the first author.
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BibTeX
@misc{huang2026sylow,
title = {{Sylow synchronization in finite groups: the good case}},
author = {Hong Yi Huang and Francesca Lisi and Aluna Rizzoli and Luca Sabatini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28403},
eprint = {2609.28403},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28403},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sylow synchronization in finite groups: the good case
AU - Hong Yi Huang
AU - Francesca Lisi
AU - Aluna Rizzoli
AU - Luca Sabatini
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28403
UR - https://arxiv.org/abs/2609.28403
AB - Let G be a finite solvable group such that, for any prime p and any quotient Q of G, there are two Sylow p-subgroups of Q intersecting in O_p(Q). Then, for every family (P_i)_i=1^n of Sylow subgroups of G for distinct primes p_1,…,p_n, there exists x ∈ G such that P_i ∩ P_i^x = O_p_i(G) for all i. This covers groups of odd order, partially settling a conjecture of the second and fourth authors and unifying old results of Bialostocki and Mann on the intersection of nilpotent subgroups. We also prove a result for all symmetric and alternating groups, completing the proof of the conjecture for simple groups as initiated by Burness and the first author.
ER -