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Some subgroup embeddings in finite groups: A mini review

In this survey paper several subgroup embedding properties related to some types of permutability are introduced and studied. | Journal of Advanced Research 2015 6 359-362 Cairo University Journal of Advanced Research MINI REVIEW Some subgroup embeddings in finite groups A mini CrossMark review A. Ballester-Bolinches a J.C. Beidleman b R. Esteban-Romero c 1 M.F. Ragland d a Departament d Algebra Universitat de Valencia Dr. Moliner 50 46100 Burjassot Valencia Spain b Department of Mathematics University of Kentucky Lexington KY 40506-0027 USA c Institut Universitari de Matematica Pura i Aplicada Universitat Politecnica de Valencia Cami de Vera s n 46022 Valencia Spain d Department of Mathematics Auburn University at Montgomery P.O. Box 244023 Montgomery AL 36124-4023 USA ARTICLE INFO ABSTRACT Article history Received 3 April 2014 Received in revised form 15 April 2014 Accepted 18 April 2014 Available online 26 April 2014 In this survey paper several subgroup embedding properties related to some types of permutability are introduced and studied. 2014 Production and hosting by Elsevier B.V. on behalf of Cairo University. Keywords Finite group Permutability S-permutability Semipermutability Primitive subgroup Quasipermutable subgroup Introduction All groups in the paper are finite. The purpose of this survey paper is to show how the embedding of certain types of subgroups of a finite group G can Corresponding author. Tel. 34 639560201 fax 34 963543918. E-mail addresses Adolfo.Ballester@uv.es A. Ballester-Bolinches clark@ms.uky.edu J.C. Beidleman Ramon.Esteban@uv.es resteban@mat.upv.es R. Esteban-Romero mragland@aum.edu M.F. Ragland . 1 Current address Departament d Algebra Universitat de Valencia Dr. Moliner 50 46100 Burjassot Valencia Spain. Peer review under responsibility of Cairo University. Production and hosting by Elsevier determine the structure of G. The types of subgroup embedding properties we consider include S-permutability S-semipermutability semipermutability primitivity and quasipermutability. A subgroup H of a group G is said to permute with a subgroup K of G if HK is a .