Maximal Subgroups of Almost Subnormal Subgroups in Division Rings

2021 ◽  
Author(s):  
Bui Xuan Hai
Keyword(s):  
Maximal Subgroups ◽  
Division Rings ◽  
Subnormal Subgroups

Subnormal Subgroups of Division Rings

1963 ◽  
Vol 15 ◽  
pp. 80-83 ◽  
Author(s):  
I. N. Herstein ◽  
W. R. Scott
Keyword(s):  
Normal Subgroup ◽  
Division Ring ◽  
Multiplicative Group ◽  
Finite Sequence ◽  
Division Rings ◽  
Subnormal Subgroups

Let K be a division ring. A subgroup H of the multiplicative group K′ of K is subnormal if there is a finite sequence (H = A0, A1, . . . , An = K′) of subgroups of K′ such that each Ai is a normal subgroup of Ai+1. It is known (2, 3) that if H is a subdivision ring of K such that H′ is subnormal in K′, then either H = K or H is in the centre Z(K) of K.

Some algebraic algebras with Engel unit groups

2019 ◽  
pp. 2150010
Author(s):  
M. H. Bien ◽  
M. Ramezan-Nassab
Keyword(s):  
Multiplicative Group ◽  
Subnormal Subgroup ◽  
Group Algebras ◽  
Algebraic Structures ◽  
Locally Finite ◽  
Division Rings ◽  
Torsion Groups ◽  
Skew Group Algebras ◽  
Subnormal Subgroups

In this paper, we study some algebras [Formula: see text] whose unit groups [Formula: see text] or subnormal subgroups of [Formula: see text] are (generalized) Engel. For example, we show that any generalized Engel subnormal subgroup of the multiplicative group of division rings with uncountable centers is central. Some of algebraic structures of Engel subnormal subgroups of the unit groups of skew group algebras over locally finite or torsion groups are also investigated.

scholarly journals On division subrings normalized by almost subnormal subgroups in division rings

2019 ◽  
Vol 80 (1) ◽  
pp. 15-27
Author(s):  
Trinh Thanh Deo ◽  
Mai Hoang Bien ◽  
Bui Xuan Hai
Keyword(s):  
Division Rings ◽  
Subnormal Subgroups

A generalized Frattini subgroup

2020 ◽  
pp. 2150026
Author(s):  
Ruslan V. Borodich
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Maximal Subgroups ◽  
General Question ◽  
Frattini Subgroup ◽  
Local Formation ◽  
Glasgow Math ◽  
A Finite Group ◽  
Subnormal Subgroups

In the work of Beidleman and Smith [On Frattini-like subgroups, Glasgow Math. J. 35 (1993) 95–98], the following question was raised: “If [Formula: see text] is a subnormal subgroup of a finite group [Formula: see text] containing [Formula: see text], then whether the supersolvability of [Formula: see text] follows the supersolvability of [Formula: see text]”. This problem was considered in works of Selkin [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)], Skiba [On the intersection of all maximal [Formula: see text]-subgroups of a finite group, Prob. Phys. Math. Tech. 3(4) (2010) 56–62], Ballester-Bolinches [On [Formula: see text]-subnormal subgroups and Frattini-like subgroups of a finite group, Glasgow Math. J. 36 (1994) 241–247] and many other authors (see monograph [Maximal Subgroups in the Theory of Classes of Finite Groups (Belaruskaya, Navuka, 1997)]). In this paper, we give the answer to the more general question: “Let [Formula: see text] be a local formation. If [Formula: see text] is a subnormal subgroup of a group [Formula: see text], then in what case [Formula: see text] will follow from [Formula: see text]”.

Maximal Subgroups of Subnormal Subgroups ofGLn(D) with Finite Conjugacy Classes

2010 ◽  
Vol 39 (1) ◽  
pp. 169-175 ◽  
Author(s):  
Dariush Kiani ◽  
Mojtaba Ramezan-Nassab
Keyword(s):  
Conjugacy Classes ◽  
Maximal Subgroups ◽  
Finite Conjugacy ◽  
Subnormal Subgroups
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