Finite nilpotent groups having exactly five conjugacy classes of non-normal subgroups are classified

Abstract

In this paper, we characterize the finite nilpotent groups having exactly five conjugacy classes of non-normal subgroups.

Keywords

• non-normal subgroup
• conjugacy class
• nilpotent group
• classification

References

1. [1] N. Blackburn, Finite Groups in which the Nonnormal Subgroups have Nontrivial Intersection, J. Algebra 3, 30–37, 1966.
2. [2] R. Brandl, Groups with few Non-normal Subgroup, Comm. Algebra 23 (6), 2091–2098, 1995.
3. [3] S.M. Chen and G.Y. Chen, A Mark of Finite Groups having exactly two Conjugacy Classes of Nonnormal Subgroups, J. Jilin Univ. Sci. 46 (6), 1097–1100, 2008.
4. [4] S.M. Chen, The Influence of Several Properties of Subgroups on the Structure of Finite Groups, PhD Thesis, Shanxi Normal University, 2009.
5. [5] L. Gong, H.P. Cao and G.Y. Chen, Finite Nilpotent Groups Having Exactly Four Conjugacy Classes of Non-normal Subgroups, Algebra Colloq. 20 (4), 579–592, 2013.
6. [6] R. La Haye and A. Rhemtulla, Groups with a bounded number of conjugacy classes of non-normal subgroups, J. Algebra 214, 41–63, 1999.
7. [7] H. Mousavi, On Finite Groups with few Nonnormal Subgroups, Comm. Algebra 27 (7), 3143–3151, 1999.
8. [8] H. Mousavi, Nilpotent groups with three conjugacy classes of non-normal subgroups, Bull. Iranian Math. Soc. 5, 1291–1300, 2014.

9. [9] J. Poland and A. Rhemtulla, The Number of Conjugacy Classes of Nonnormal Subgroups in Nilpotent Groups, Comm. Algebra 24 (10), 3237–3245, 1996.
10. [10] D.J.S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, 1982.
11. [11] G. Walls, Trivial Intersection Groups, Arch. Math. 32, 1–4, 1979.