Certain structural theorems on subnormal subgroups of $GL_n(D)$
Khanh, Huynh Viet ; Hai, Bui Xuan
arXiv, Tome 2019 (2019) no. 0, / Harvested from
Let $D$ be a non-commutative division ring with center $F$, and $G$ a subnormal subgroup of $GL_n(D)$. Assume additionally that $D$ contains at least $4$ elements in the case $n>1$. We show that if $G$ is locally solvable, then it is contained in $F$. This fact generalizes the results of Stuth and Huzurbazar which asserted that any solvable or locally nilpotent subnormal subgroup of $D^*$ is central. Also, if $G$ contains a non-abelian solvable-by-finite maximal subgroup, then $[D:F]<\infty$, and there exists a maximal subfield $K$ of $M_n(D)$ such that $K/F$ is a finite Galois extension. We also address maximal subgroups of $G$ of some special types such as solvable, polycyclic-by-finite subgroups.
Publié le : 2019-03-23
Classification:  Mathematics - Rings and Algebras 
@article{1903.10868,
     author = {Khanh, Huynh Viet and Hai, Bui Xuan},
     title = {Certain structural theorems on subnormal subgroups of $GL\_n(D)$},
     journal = {arXiv},
     volume = {2019},
     number = {0},
     year = {2019},
     language = {en},
     url = {http://dml.mathdoc.fr/item/1903.10868}
}

Khanh, Huynh Viet; Hai, Bui Xuan. Certain structural theorems on subnormal subgroups of $GL_n(D)$. arXiv, Tome 2019 (2019) no. 0, . http://gdmltest.u-ga.fr/item/1903.10868/