Irreducible disconnected systems in groups

LetG be an arbitrary group with a subgroupA. The subdegrees of (A, G) are the indices A:A ∪A ⁹] (wheregεG). Equivalent definitions of that concept are given in IP] and K]. IfA is not normal inG and all the subdegrees of (A, G) are finite, we attach to (A, G) the common divisor graph Γ: its vertices are the non-unit subdegrees of (A, G), and two different subdegrees are joined by an edge iff they arenot coprime. It is proved in IP] that Γ has at most two connected components. Assume that Γ is disconnected. LetD denote the subdegree set of (A, G) and letD ₁ be the set of all the subdegrees in the component of Γ containing min(D−{1}). We proved K, Theorem A] that ifA is stable inG (a property which holds whenA or G:A] is finite), then the setH={g ε G| A:A ∪A ^g ] εD ₁ ∪ {1}} is a subgroup ofG. In this case we say thatA<H<G is a disconnected system (briefly: a system). In the current paper we deal with some fundamental types of systems. A systemA<H<G is irreducible if there does not exist 1<N△G such thatAN<H andAN/N<H/N<G/N is a system. Theorem A gives restrictions on the finite nilpotent normal subgroups ofG, whenG possesses an irreducible system. In particular, ifG is finite then Fit(G) is aq-group for a certain primeq. We deal also with general systems. Corollary (4.2) gives information about the structure of a finite groupG which possesses a system. Theorem B says that for any systemA<H<G,N _G (N _G (A))=N _G (A). Theorem C and Corollary C’ generalize a result of Praeger P, Theorem 2]. The content of this paper corresponds to a part of the author’s Ph.D. thesis carried out at Tel Aviv University under the supervision of Prof. Marcel Herzog.