Standard

Maximal and Submaximal x-Subgroups. / Guo, W.; Revin, D. O.

In: Algebra and Logic, Vol. 57, No. 1, 19.05.2018, p. 9-28.

Research output: Contribution to journal › Article › peer-review

Harvard

Guo, W & Revin, DO 2018, 'Maximal and Submaximal x-Subgroups', Algebra and Logic, vol. 57, no. 1, pp. 9-28. https://doi.org/10.1007/s10469-018-9475-8

APA

Guo, W., & Revin, D. O. (2018). Maximal and Submaximal x-Subgroups. Algebra and Logic, 57(1), 9-28. https://doi.org/10.1007/s10469-018-9475-8

Vancouver

Guo W, Revin DO. Maximal and Submaximal x-Subgroups. Algebra and Logic. 2018 May 19;57(1):9-28. doi: 10.1007/s10469-018-9475-8

Author

Guo, W. ; Revin, D. O. / Maximal and Submaximal x-Subgroups. In: Algebra and Logic. 2018 ; Vol. 57, No. 1. pp. 9-28.

BibTeX

@article{e688d8dc404545218b7fc08ae642b517,
title = "Maximal and Submaximal x-Subgroups",
abstract = "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{\textquoteright}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? ",
keywords = "Dπ-property, finite group, Hall π-subgroup, maximal {\dh}”›-subgroup, submaximal {\dh}”›-subgroup, maximal x subgroup, submaximal x-subgroup, D-pi-pproperty, maximal x-subgroup, CLASSIFICATION, Hall pi-subgroup, CONJECTURE, PRIMITIVE PERMUTATION-GROUPS, FINITE-GROUPS, HALL SUBGROUPS, PI, ODD INDEX, PRONORMALITY",
author = "W. Guo and Revin, {D. O.}",
year = "2018",
month = may,
day = "19",
doi = "10.1007/s10469-018-9475-8",
language = "English",
volume = "57",
pages = "9--28",
journal = "Algebra and Logic",
issn = "0002-5232",
publisher = "Springer US",
number = "1",

}

RIS

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 -

ID: 13487944