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 boolean OR(g is an element of G) Ax)(g) = G - {1(G)} for all x is an element of G - A. gEG 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. (C) 2015 Elsevier Inc. All rights reserved.