阿贝尔群被超-(循环或有限)群的可裂扩张(Ⅰ)

设F是由f(p)所局部定义的可解群系,G∈F,A是ZG-模.我们称A的一个p-主因子U/V在G中是F-中心的,如果G/CG(U/V)∈f(p).否则称U/V在G中是非中心的.本文证明了:设G是超-(有限或循环)的局部可解群,A是Artinian ZG-模且所有的不可约ZG-因子都是有限的;F为由f(p)所局部定义的局部可解群系,且对任意的p∈π,f(p)≠φ,f(∞) f(p).如果G∈F,且A的所有不可约ZG-因子在G中均是F-非中心的,则A被G的扩张在A上共轭可裂..

Splitting extensions of Abelian by hyper-(cyclic or finite) groups (Ⅰ)

Let F be a formation locally defined by f(p), G∈ F and A a ZG-module, where p∈π= {all primes and ∞}. Then a p-main-factor U/V of is said to be F-central in G, if G/CG(U/V)∈f(p). Otherwise, it is said to be F-eccentric in G. In this paper, the following results are proved: Let F be a locally defined formation consisting of locally soluble groups, G a hyper-(cyclic or finite) locally soluble group and A an artinian ZG-module with all irreducible ZG-factors of A being finite, if G∈F, f(∞) f(p),f(p)≠φ for each p∈π and all irreducible ZG-factors of A are F-eccentric in G, then any extension E of A by G splits conjugately over A.