TY - JOUR

T1 - Conjugacy of Maximal and Submaximal 픛-Subgroups

AU - Guo, W.

AU - Revin, D. O.

N1 - Publisher Copyright: © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2018/7/1

Y1 - 2018/7/1

N2 - Let 픛 be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup H of a finite group G a submaximal 픛-subgroup if there exists an isomorpic embedding ϕ: G ↪ G* of the group G into some finite group G* under which Gϕ is subnormal in G* and Hϕ = K ∩Gϕ for some maximal 픛-subgroup K of G*. We discuss the following question formulated by Wielandt: Is it always the case that all submaximal 픛-subgroups are conjugate in a finite group G in which all maximal 픛-subgroups are conjugate? This question strengthens Wielandt’s known problem of closedness for the class of [InlineMediaObject not available: see fulltext.]-groups under extensions, which was solved some time ago. We prove that it is sufficient to answer the question mentioned for the case where G is a simple group.

AB - Let 픛 be a class of finite groups closed under taking subgroups, homomorphic images, and extensions. Following H. Wielandt, we call a subgroup H of a finite group G a submaximal 픛-subgroup if there exists an isomorpic embedding ϕ: G ↪ G* of the group G into some finite group G* under which Gϕ is subnormal in G* and Hϕ = K ∩Gϕ for some maximal 픛-subgroup K of G*. We discuss the following question formulated by Wielandt: Is it always the case that all submaximal 픛-subgroups are conjugate in a finite group G in which all maximal 픛-subgroups are conjugate? This question strengthens Wielandt’s known problem of closedness for the class of [InlineMediaObject not available: see fulltext.]-groups under extensions, which was solved some time ago. We prove that it is sufficient to answer the question mentioned for the case where G is a simple group.

KW - finite group

KW - Hall π-subgroup

KW - maximal 𝔛-subgroup

KW - submaximal 𝔛-subgroup

KW - [InlineMediaObject not available: see fulltext.]-property

KW - Hall p-subgroup

KW - Dpproperty

KW - DX-property.

KW - maximal X-subgroup

KW - HALL SUBGROUPS

KW - FINITE-GROUPS

KW - submaximal X-subgroup

UR - http://www.scopus.com/inward/record.url?scp=85054182700&partnerID=8YFLogxK

U2 - 10.1007/s10469-018-9490-9

DO - 10.1007/s10469-018-9490-9

M3 - Article

AN - SCOPUS:85054182700

VL - 57

SP - 169

EP - 181

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 3

ER -