An approach to Quillen's conjecture via centralisers of simple groups

• Autores: Piterman, Kevin
• Año de publicación: 2021
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: p-subgroups; finite posets; quillen's conjecture; simple groups
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

For any given subgroup H of a finite group G, the Quillen poset of nontrivial elementary abelian p-subgroups is obtained from by attaching elements via their centralisers in H. We exploit this idea to study Quillen's conjecture, which asserts that if is contractible then G has a nontrivial normal p-subgroup. We prove that the original conjecture is equivalent to the-Acyclic version of the conjecture (obtained by replacing 'contractible' by '-Acyclic'). We also work with the-Acyclic (strong) version of the conjecture, reducing its study to extensions of direct products of simple groups of p-rank at least. This allows us to extend results of Aschbacher and Smith and to establish the strong conjecture for groups of p-rank at most.
Fil: Piterman, Kevin. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina