{"title":"Shunkov Groups Saturated with Almost Simple Groups","authors":"N. V. Maslova, A. A. Shlepkin","doi":"10.1007/s10469-023-09725-y","DOIUrl":null,"url":null,"abstract":"<p>A group <i>G</i> is called a Shunkov group (a conjugate biprimitive finite group) if, for any of its finite subgroups <i>H</i> in the factor group <i>N</i><sub><i>G</i></sub>(<i>H</i>)/<i>H</i>, every two conjugate elements of prime order generate a finite subgroup. We say that a group is saturated with groups from the set 𝔐 if any finite subgroup of the given group is contained in its subgroup isomorphic to some group in 𝔐. We show that a Shunkov group <i>G</i> which is saturated with groups from the set 𝔐 possessing specific properties, and contains an involution <i>z</i> with the property that the centralizer <i>C</i><sub><i>G</i></sub>(<i>z</i>) has only finitely many elements of finite order will have a periodic part isomorphic to one of the groups in 𝔐. In particular, a Shunkov group <i>G</i> that is saturated with finite almost simple groups and contains an involution <i>z</i> with the property that the centralizer <i>C</i><sub><i>G</i></sub>(<i>z</i>) has only finitely many elements of finite order will have a periodic part isomorphic to a finite almost simple group.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2023-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra and Logic","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s10469-023-09725-y","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
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Abstract
A group G is called a Shunkov group (a conjugate biprimitive finite group) if, for any of its finite subgroups H in the factor group NG(H)/H, every two conjugate elements of prime order generate a finite subgroup. We say that a group is saturated with groups from the set 𝔐 if any finite subgroup of the given group is contained in its subgroup isomorphic to some group in 𝔐. We show that a Shunkov group G which is saturated with groups from the set 𝔐 possessing specific properties, and contains an involution z with the property that the centralizer CG(z) has only finitely many elements of finite order will have a periodic part isomorphic to one of the groups in 𝔐. In particular, a Shunkov group G that is saturated with finite almost simple groups and contains an involution z with the property that the centralizer CG(z) has only finitely many elements of finite order will have a periodic part isomorphic to a finite almost simple group.
期刊介绍：
This bimonthly journal publishes results of the latest research in the areas of modern general algebra and of logic considered primarily from an algebraic viewpoint. The algebraic papers, constituting the major part of the contents, are concerned with studies in such fields as ordered, almost torsion-free, nilpotent, and metabelian groups; isomorphism rings; Lie algebras; Frattini subgroups; and clusters of algebras. In the area of logic, the periodical covers such topics as hierarchical sets, logical automata, and recursive functions.
Algebra and Logic is a translation of ALGEBRA I LOGIKA, a publication of the Siberian Fund for Algebra and Logic and the Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences.
All articles are peer-reviewed.
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