Year: 2024
Type: article
Abstract: Building on previous work by Caicedo and the second author, we develop a method that decides the existence of units of finite order in blocks of $$\mathbb {Z}_p G$$ Z p G of defect 1. This allows us to prove that if p is a prime and G is a finite group whose Sylow p-subgroup has order p, then any unit u of $$\mathbb {Z}G$$ Z G of order p is conjugate to an element of $$\pm G$$ ± G within $$\mathbb {Q}G$$ Q G . This is a special case of the Zassenhaus conjecture. We also prove some new results on units of finite order in $$\mathbb {Z}{{\text {PSL}}}(2,q)$$ Z PSL ( 2 , q ) for certain q, and construct a unit of order 15 in $$V(\mathbb {Z}_{(3,5)}{{\text {PSL}}}(2,16))$$ V ( Z ( 3 , 5 ) PSL ( 2 , 16 ) ) which is a 3- and 5-local counterexample to the Zassenhaus conjecture, raising the hope that our methods may lead to a global counterexample among simple groups.
Source: Revista Matemática Complutense
Authors: Florian Eisele, Leo Margolis
Institutions: University of Manchester, Institute of Mathematical Sciences
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $2,990
Awards: G1175115578, G121809054, G1701110065, G1855990336, G2262748287, G2281333282, G2309174360, G2660406951, G279363373, G2967504892, Representation theory over local rings, G300979063, G34346518, G3480869486, G3490997933, G3518538256, G4233400527, G425668148, G430777504, G4526231891, G5080475149, G5394525568, G5421974903, G5845768719, G6200043914, G7084143925, G897484643, G99296377
OpenAlex ID: W4405348779