On locally normal Fitting classes of finite soluble groups

✍ Scribed by Stephanie Reifferscheid

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 219 KB
• Volume: 261
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(02)00561-6

✦ Synopsis

Let X, F, X ⊆ F, be non-trivial Fitting classes of finite soluble groups such that G X is an X-injector of G for all G ∈ F. Then X is said to be normal in F (F-normal). We show that for a subgroup-closed Fitting class X the collection of all subgroup-closed Fitting classes in which X is normal forms a complete, distributive and atomic lattice. Moreover, X is determined uniquely by the unique maximal subgroup-closed Fitting class in which X is normal, and in many cases there is an algorithm to describe this class explicitly. Further we show that if F is a subgroup-closed Fitting class such that a unique minimal subgroup-closed F-normal Fitting class exists, the collection of all subgroup-closed F-normal Fitting classes also forms a complete distributive lattice which in addition is dual atomic if F is of bounded nilpotent length.