TY - JOUR

T1 - Classification and properties of the π -submaximal subgroups in minimal nonsolvable groups

AU - Guo, Wenbin

AU - Revin, Danila O.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - Let π be a set of primes. According to H. Wielandt, a subgroup H of a finite group X is called a π-submaximal subgroup if there is a monomorphism ϕ: X→ Y into a finite group Y such that Xϕ is subnormal in Y and Hϕ= K∩ Xϕ for a π-maximal subgroup K of Y. In his talk at the celebrated conference on finite groups in Santa-Cruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the π-submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the intravariancy, the conjugacy in the automorphism group etc. In the article, for every set π of primes, we obtain a description of the π-submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandt’s problem.

AB - Let π be a set of primes. According to H. Wielandt, a subgroup H of a finite group X is called a π-submaximal subgroup if there is a monomorphism ϕ: X→ Y into a finite group Y such that Xϕ is subnormal in Y and Hϕ= K∩ Xϕ for a π-maximal subgroup K of Y. In his talk at the celebrated conference on finite groups in Santa-Cruz (USA) in 1979, Wielandt posed a series of open questions and among them the following problem: to describe the π-submaximal subgroup of the minimal nonsolvable groups and to study properties of such subgroups: the pronormality, the intravariancy, the conjugacy in the automorphism group etc. In the article, for every set π of primes, we obtain a description of the π-submaximal subgroup in minimal nonsolvable groups and investigate their properties, so we give a solution of Wielandt’s problem.

KW - Minimal nonsolvable group

KW - Minimal simple group

KW - Pronormal subgroup

KW - π-Maximal subgroup

KW - π-Submaximal subgroup

KW - FINITE SIMPLE-GROUPS

KW - EXISTENCE

KW - pi-Maximal subgroup

KW - THEOREM

KW - CONJECTURE

KW - SYLOW TYPE

KW - HALL SUBGROUPS

KW - PRONORMALITY

KW - CRITERION

KW - pi-Submaximal subgroup

UR - http://www.scopus.com/inward/record.url?scp=85047889972&partnerID=8YFLogxK

U2 - 10.1007/s13373-017-0112-y

DO - 10.1007/s13373-017-0112-y

M3 - Article

AN - SCOPUS:85047889972

VL - 8

SP - 325

EP - 351

JO - Bulletin of Mathematical Sciences

JF - Bulletin of Mathematical Sciences

SN - 1664-3607

IS - 2

ER -