2025/01/20 by Hamid Mousavi, Mousavi, Hamid
Engineering · #20D05 #20D99 #Advanced Control Systems Optimization #FOS: Mathematics #Group Theory (math.GR) #Process Optimization and Integration
paper · pdf · doi:10.48550/arxiv.2501.11486
openalex publication_date 2025/01/20 · openalex created_date 2025/01/24 · openalex updated_date 2026/07/28
Let G be a finite group and x be an element of G. Define \textrmSolG(x) as the set of all y ∈ G such that ⟨ x,y⟩ is soluble. We provide an equivalent condition for the normalizer-solubilizer conjecture, namely |NG(⟨ x⟩)| | |\textrmSolG(x)|, where NG(⟨ x⟩) is the normalizer of ⟨ x⟩. Furthermore, we demonstrate that the conjecture holds in the special case where NG(⟨ x⟩) is a Frobenius group with kernel CG(x), the centralizer of x, and |NG(⟨ x⟩): CG(x)| is of prime order. Finally, we will classify all finite simple groups G that contain an element x for which \textrmSolG(x) is a maximal subgroup of order pq, where p and q are prime numbers.