# A note on solvable maximal subgroups in subnormal subgroups of ${\mathrm GL}_n(D)$

@article{Khanh2018ANO, title={A note on solvable maximal subgroups in subnormal subgroups of \$\{\mathrm GL\}\_n(D)\$}, author={Huynh Viet Khanh and Bui Xuan Hai}, journal={arXiv: Rings and Algebras}, year={2018} }

Let $D$ be a non-commutative division ring, $G$ a subnormal subgroup of ${\mathrm GL}_n(D)$. In this note we show that if $G$ contains a non-abelian solvable maximal subgroup, then $n=1$ and $D$ is a cyclic algebra of prime degree over $F$.

## One Citation

On almost subnormal subgroups in division rings

- Mathematics
- 2019

Let $D$ be a division ring with infinite center $F$, and $G$ an almost subnormal subgroup of $D^*$. In this paper, we show that if $G$ is locally solvable, then $G\subseteq F$. Also, assume that $M$…

## References

SHOWING 1-10 OF 16 REFERENCES

SOLUBLE MAXIMAL SUBGROUPS IN GLn(D)

- Mathematics
- 2011

Let D be an F-central non-commutative division ring. Here, it is proved that if GLn(D) contains a non-abelian soluble maximal subgroup, then n = 1, [D : F] < ∞, and D is cyclic of degree p, a prime.…

Polynomial identities and maximal subgroups of skew linear groups

- Mathematics
- 2007

Suppose that D is a division ring with center F and N is a non-central normal subgroup of GLn(D). In this paper we generalize some known results about maximal subgroups of GLn(D) to maximal subgroups…

Finitely Generated Subnormal Subgroups of GLn(D) Are Central

- Mathematics
- 2000

Abstract Let D be an infinite division algebra of finite dimension over its center. Assume that N is a subnormal subgroup of GLn(D) with n ≥ 1. It is shown that if N is finitely generated, then N is…

Normalizers of nilpotent subgroups of division rings

- Mathematics
- 2007

Let D be an arbitrary division ring and G a nilpotent subgroup of the multiplicative group D* of D of class at most 2 such that D is generated as a division ring by G and the centralizer in D of G.…

Nilpotent and polycyclic-by-finite maximal subgroups of skew linear groups

- Mathematics
- 2013

Let D be an infinite division ring, n a natural number and N a subnormal subgroup of GLn(D) such that n = 1 or the center of D contains at least five elements. This paper contains two main results.…

Maximal subgroups of GLn(D)

- Mathematics
- 2003

Abstract In this paper we study the structure of locally solvable, solvable, locally nilpotent, and nilpotent maximal subgroups of skew linear groups. In [S. Akbari et al., J. Algebra 217 (1999)…

Free subgroups in maximal subgroups of skew linear groups

- Mathematics, Computer ScienceInt. J. Algebra Comput.
- 2019

The study of the existence of free groups in skew linear groups have begun since the last decades of the 20th century. The starting point is the theorem of Tits (1972), now often referred to as Tits’…

Solvable subgroups in prime rings

- Mathematics
- 1981

Let R be a prime ring with center Z and group of units U. The main theorem shows that any solvable normal subgroups of U must lie in Z, provided that R is not a domain, Z is large enough, and that…

Soluble normal subgroups of skew linear groups

- Mathematics
- 1986

A skew linear group is a subgroup of GL(n,D) for some positive integer n and division ring D. In [13] and [14] we studied a locally finite normal subgroup H of a skew linear group G and found in…

The algebraic structure of group rings

- Mathematics
- 1977

There appeared in 1976 an expository paper by the present author [52] entitled "What is a group ringV This question, rhetorical as it is, may nevertheless be answered directly by saying that for a…