On Groups Admitting a Disconnected Common Divisor Graph

Abstract

LetGbe an arbitrary group with a subgroupA. Each double cosetAgAis a union of right cosetsAu. The cardinality of the set {Au∣u∈G,Au⊆AgA} is called asubdegreeof (A,G) and is denoted by [AgA:A]. Thus for each double cosetAgAwe have a corresponding subdegree. An equivalent definition of the subdegree concept is given in [2]. IfAis not normal inGand all the subdegrees of (A,G) are finite, we attach to (A,G) thecommon divisor graphΓ: its vertices are the nonunit subdegrees of (A,G), and two different subdegrees are joined by an edge iff they arenotcoprime. It is proved in [2] that Γ has at most two connected components. We prove that if Γ is disconnected andAsatisfies a certain “regularity” property (a property which holds whenAor [G:A] is finite, and is called in this paper stability), thenGhas a nice structure. To be more precise, letDdenote the subdegree set of (A,G) and letD1be the set of all the subdegrees in the connected component of Γ containing min(D−{1}). Then we prove (Theorem A) that the setH=⋃[AgA:A]∈D1∪{1}AgAis a subgroup ofGandNG(A)<H<G. Some interesting properties of the subgroupHare described in Theorem B. Theorems D and E describe some properties of the subdegrees in the disconnected case.