The influence of the nilpotentlizers on group structur

2024/02/24 by N. Ahmadkhah, Ahmadkhah, N., Mohammad Zarrin +1
Mathematics · #20D10 #20D15 #20D60 #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2402.15916

openalex publication_date 2024/02/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a finite group G and an element x∈ G, the subset nil_G(x)=\y∈ G | ~~ is ~~ nilpotent\ is called nilpotentizer of x in G. In this paper, we give two solvabilty criteria for a finite group by the structure and the size of nilpotentizer of an element on finite group. In fact, we show that if there exists an element x of G such that nil_G(x) generates a maximal subgroup of G and the simple commutator of weight 2 ~~or ~~3 of elements of nil_G(x) is equal to 1 or |nil_G(x)|= pⁿ, where p is prime and n=1, 2. Then G is a solvable group.

Related

• A nilpotency criterion for finite groups
• Solvability of a group based on its number of subgroups
• p-sylowizers and p-nilpotency of finite groups
• Criteria for solubility and nilpotency of finite groups with automorphisms
• Variations on the Thompson theorem