On groups admitting a disconnected common divisor graph

[摘要] Let G be an arbitrary group with a subgroup A, Each double coset AgA is a union of right cosets Au. The cardinality of the set {Au \ u is an element of G, Au subset of or equal to AgA} is called a subdegree of (A, G) and is denoted by [AgA : A]. Thus for each double coset AgA we have a corresponding subdegree. An equivalent definition of the subdegree concept is given in [2]. If A is not normal in G and all the subdegrees of (A, G) are finite, we attach to (A, G) the common divisor graph Gamma: its vertices are the nonunit subdegrees of (A, G), and two different subdegrees are joined by an edge iff they are not coprime. It is proved in [2] that Gamma has at most two connected components. We prove that if Gamma is disconnected and A satisfies a certain ''regularity'' property (a property which holds when A or [G: A] is finite, and is called in this paper stability), then G has a nice structure. To be more precise, let D denote the subdegree set of (A, G) and let D-1 be the set of all the subdegrees in the connected component of Gamma containing min(D- {1}). Then we prove (Theorem A) that the set H = U ([AgA:A]is an element of D1 boolean OR{1})AgA is a subgroup of G and N-G(A) < H< G. Some interesting properties of the subgroup H are described in Theorem B. Theorems D and E describe some properties of the subdegrees in the disconnected case. (C) 1997 Academic Press.

[发布日期] 1997-07-15