Covering a Finite Group by the Conjugates of a Coset

We study pairs (G,A) where G is a finite group and A < G is maximal, satisfying ⋃ g∈G (Ax) = G − {1G} for all x ∈ G − A. We prove that this condition defines a class of permutation groups, denoted CCI, which is a subclass of the class of primitive permutation groups. We prove that CCI contains the class of 2-transitive groups. We also prove that groups in CCI are either affi ne or almost simple. In the affi ne case each CCI group must be 2-transitive, while an almost simple CCI group needs not be 2transitive. We give various results on the almost simple case and compare between the CCI class and other recently studied classes of groups which lie between the 2-transitive and the primitive permutation groups.