A generalization of the Arad--Ward theorem on Hall subgroups

2025/01/10 by Yang, N., Buturlakin, A. A.
#20D20 #20D30 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2501.05865

Abstract

For a set of primes π, denote by E_π the class of finite groups containing a Hall π-subgroup. We establish that E_π1∩ E_π2 is contained in E_π1∩π2. As a corollary, we prove that if π is a set of primes, l is an integer such that 2\leqslant l<|π| and G is a finite group that contains a Hall ρ-subgroup for every subset ρ of π of size l, then G contains a solvable Hall π-subgroup.

Related

• A Criterion for the Existence of a Solvable π-Hall Subgroup in a Finite Group
• On the converse of Hall's theorem
• Frattini Argument for Hall subgroups
• Finite groups with permutable Hall subgroups
• On the number of conjugacy classes of π-elements in finite groups