{"title":"可溶性基团的效价","authors":"B. A. F. Wehrfritz","doi":"10.1007/s00605-023-01897-0","DOIUrl":null,"url":null,"abstract":"Abstract We prove in particular that if G is a soluble group with no non-trivial locally finite normal subgroups, then G is p-potent for every prime p for which G has no Prüfer p-sections. (A group G is p-potent if for every power n of p and for any element x of G of infinite order or of finite order divisible by n there is a normal subgroup N of G of finite index such that the order of x modulo N is n. A Prüfer p-group is an infinite locally cyclic p-group.) This extends to soluble groups in general, and gives a more direct proof of, recent results of Azarov on polycyclic groups and soluble minimax groups.","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":"15 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Potency in soluble groups\",\"authors\":\"B. A. F. Wehrfritz\",\"doi\":\"10.1007/s00605-023-01897-0\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract We prove in particular that if G is a soluble group with no non-trivial locally finite normal subgroups, then G is p-potent for every prime p for which G has no Prüfer p-sections. (A group G is p-potent if for every power n of p and for any element x of G of infinite order or of finite order divisible by n there is a normal subgroup N of G of finite index such that the order of x modulo N is n. A Prüfer p-group is an infinite locally cyclic p-group.) This extends to soluble groups in general, and gives a more direct proof of, recent results of Azarov on polycyclic groups and soluble minimax groups.\",\"PeriodicalId\":18913,\"journal\":{\"name\":\"Monatshefte für Mathematik\",\"volume\":\"15 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-10-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Monatshefte für Mathematik\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s00605-023-01897-0\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-023-01897-0","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Abstract We prove in particular that if G is a soluble group with no non-trivial locally finite normal subgroups, then G is p-potent for every prime p for which G has no Prüfer p-sections. (A group G is p-potent if for every power n of p and for any element x of G of infinite order or of finite order divisible by n there is a normal subgroup N of G of finite index such that the order of x modulo N is n. A Prüfer p-group is an infinite locally cyclic p-group.) This extends to soluble groups in general, and gives a more direct proof of, recent results of Azarov on polycyclic groups and soluble minimax groups.