Acta Mathematica Sinica, English Series Apr., 2011, Vol. 27, No. 4, pp. 715–724 Published online: March 15, 2011 DOI: 10.1007/s10114-011-9310-9 Http://www.ActaMath.com Acta Mathematica Sinica, English Series © Springer-Verlag Berlin Heidelberg & The Editorial Office of AMS 2011 Finite Groups with Some Pronormal Subgroups Zhen Cai SHEN School of Mathematical Sciences, Suzhou University, Suzhou 215006, P. R. China E-mail : zhencai 688@sina.com Wu Jie SHI 1) School of Mathematics and Statistics, Chongqing University of Arts and Sciences, Chongqing 402160, P. R. China E-mail : wjshi@suda.edu.cn Abstract A subgroup H of finite group G is called pronormal in G if for every element x of G, H is conjugate to H x in H, H x . A finite group G is called P RN -group if every cyclic subgroup of G of prime order or order 4 is pronormal in G. In this paper, we find all P RN -groups and classify minimal non-P RN -groups (non-P RN -group all of whose proper subgroups are P RN -groups). At the end of the paper, we also classify the finite group G, all of whose second maximal subgroups are P RN -groups. Keywords Pronormal subgroups, P RN -groups, minimal non-P RN -groups, PN -groups, minimal subgroups, p-nilpotent groups MR(2000) Subject Classification 20D10, 20D20 1 Introduction All groups considered in this paper will be finite. A group is called a PN -group if its minimal subgroups are normal. In 1970, Buckley [1] proved that a PN -group of odd order is super- solvable. In 1980, the structure of non-PN -group whose proper subgroups is PN -groups has been described by Sastry [2]. The PN -groups were generalized by many authors. In 1988, Asaad [3] said that a group G is an (A)-group if every subgroup of G of prime order is pronor- mal in G and either the Sylow 2-subgroups of G are abelian or every cyclic subgroup of G of order 4 is pronormal in G, and investigated finite simple groups, all of whose second maximal subgroups are (A)-groups. In 1998, Li [4] called a subgroup X of group G an NE-subgroup if X = N G (X) ∩ X G , where X G is the normal closure of X in G and defined PE-groups as the groups in which every minimal subgroup is an NE-subgroup. At the same time, the structures of PE-groups and minimal non-PE-groups (a non-PE-group each of whose proper subgroups is a PE-group) have been classified. In the paper, we study other generalizations of PN -groups and give the following: Received June 3, 2009, revised June 8, 2010, accepted July 23, 2010 Supported by Natural Science Foundation of China (Grant No. 10871032), Graduate Student Research and Innovation Program of Jiangsu Province (Grant No. CX10B - 028Z) 1) Corresponding author