Abstract
We study pairs (G,A) where G is a finite group and A<G is maximal, satisfying ⋃g∈G(Ax)g=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 affine or almost simple. In the affine case each CCI group must be 2-transitive, while an almost simple CCI group needs not be 2-transitive. 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.
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