关于有限群的S-半置换子群的一点注记.pdfVIP

第2O卷 第2期 湖 南 文 理 学 院 学 报 (自然 科 学 版) Vl01．2ONo．2 2008年6月 Journal of Hunan University of Arts and Science(Natural Science Edition) Jun．2008 文章编号：1672—6146(2008)02—0003—03 A note on S·semipermutable subgroups of finite groups TANG Zeng—lin (Dept．of Math，Human University of Arts and Science，Changde，Hunan，41 5000) Abstract： By s—semipermutable maximal subgroups of G／(Ⅳ1nⅣ2)∈F．A formation F is said to be saturated non-cyclic Sylow subgroups to characterize the structure of if G／ (G)∈ implies G∈F．In this paper,all groups finite groups，the results were obtained as follows：Let G be a are finite；U demotes the class of all super-solvable finite group and F a saturated formation containing U．Suppose grouP s． Ghas a solvablenorm al subgroupH suchthatG／HEF．Ifevery 1 Preliminaries maximal subgroup of non—cyclic Sylow subgroups of F( is s—semipermutable in G．then G ∈F． Lemma 1【 Let Gbe afinite group． Key words：finite group；maximal subgroup；cyclic subgroup． (1)Assume H is a s—semipermutable subgroup of Cl C nudaber：O 152．1 · Document Code：A G，ifHSK GthenHis s—semiperm utable inK． (2)Assume H is a s—semipermutable subgroup of Recall that two subgroups H and K of a group G G and Jvq G．then HN／Nis s—semipermutable in G／N． are said to be permute if HK=KH．A subgroup of G is Lemma 2【 Let G be a group and F a saturated called quasinorm al in G if it permutes with every form ation containing the class of supersolvable groups． subgroup of G．A subgroup of G is called s— If there exists a norm al cyclic subg