TY - JOUR

T1 - Maximal and Submaximal x-Subgroups

AU - Guo, W.

AU - Revin, D. O.

PY - 2018/5/19

Y1 - 2018/5/19

N2 - Let x 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 x-subgroup if there exists an isomorphic embedding ϕ : G ↪ G * of G into some finite group G * under which G ϕ is subnormal in G * and H ϕ = K ∩ G ϕ for some maximal x-subgroup K of G *. In the case where x coincides with the class of all π-groups for some set π of prime numbers, submaximal x-subgroups are called submaximal π-subgroups. In his talk at the well-known conference on finite groups in Santa Cruz in 1979, Wielandt emphasized the importance of studying submaximal π-subgroups, listed (without proof) certain of their properties, and formulated a number of open questions regarding these subgroups. Here we prove properties of maximal and submaximal x- and π-subgroups and discuss some open questions both Wielandt’s and new ones. One of such questions due to Wielandt reads as follows: Is it always the case that all submaximal x-subgroups are conjugate in a finite group G in which all maximal x-subgroups are conjugate?

AB - Let x 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 x-subgroup if there exists an isomorphic embedding ϕ : G ↪ G * of G into some finite group G * under which G ϕ is subnormal in G * and H ϕ = K ∩ G ϕ for some maximal x-subgroup K of G *. In the case where x coincides with the class of all π-groups for some set π of prime numbers, submaximal x-subgroups are called submaximal π-subgroups. In his talk at the well-known conference on finite groups in Santa Cruz in 1979, Wielandt emphasized the importance of studying submaximal π-subgroups, listed (without proof) certain of their properties, and formulated a number of open questions regarding these subgroups. Here we prove properties of maximal and submaximal x- and π-subgroups and discuss some open questions both Wielandt’s and new ones. One of such questions due to Wielandt reads as follows: Is it always the case that all submaximal x-subgroups are conjugate in a finite group G in which all maximal x-subgroups are conjugate?

KW - Dπ-property

KW - finite group

KW - Hall π-subgroup

KW - maximal 𝔛-subgroup

KW - submaximal 𝔛-subgroup

KW - maximal x subgroup

KW - submaximal x-subgroup

KW - D-pi-pproperty

KW - maximal x-subgroup

KW - CLASSIFICATION

KW - Hall pi-subgroup

KW - CONJECTURE

KW - PRIMITIVE PERMUTATION-GROUPS

KW - FINITE-GROUPS

KW - HALL SUBGROUPS

KW - PI

KW - ODD INDEX

KW - PRONORMALITY

UR - http://www.scopus.com/inward/record.url?scp=85047159923&partnerID=8YFLogxK

U2 - 10.1007/s10469-018-9475-8

DO - 10.1007/s10469-018-9475-8

M3 - Article

AN - SCOPUS:85047159923

VL - 57

SP - 9

EP - 28

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 1

ER -