Number Of Elements Outside Any Conjugate Of A Subgroup

Let G be a finite group,and H be a proper subgroup.In this thesis,we consider the number of elements of G that do not lie in any conjugate of H,and explore the equivalent conditions such that the number of elements of G that do not lie in any conjugate of H is exactly|H|.We prove that a subgroup such that the number of elements outside any conjugate of it is exactly the order of the subgroup is a maximal subgroup.At the same time.we depict finite groups whose nontrivial subgroups all satisfy the condition that the elements of G that do not lie in any conjugate of a subgroup is exactly the order of the subgroup itself.For symmetric or nilpotent groups,we try to find all of its maximal subgroups satisfying the number of elements of G outside any conjugate of a maximal subgroup is exactly the order of the maximal subgroup. We also have a extension of Zipper Lemma.