[Paper Review] A note on solvable maximal subgroups in subnormal subgroups of ${\mathrm GL}_n(D)$

H. V. Khanh, Bui Xuan Hai|arXiv (Cornell University)|Sep 2, 2018

Finite Group Theory Research12 references4 citations

TL;DR

This paper proves that if a subnormal subgroup $ G $ of $ \mathrm{GL}_n(D) $, where $ D $ is a non-commutative division ring with center $ F $, contains a non-abelian solvable maximal subgroup, then necessarily $ n = 1 $ and $ D $ is a cyclic algebra of prime degree over $ F $. The result confirms a special case of a conjecture on maximal subgroups in skew linear groups, showing such subgroups cannot exist in $ \mathrm{GL}_n(D) $ for $ n > 1 $, and characterizes the structure of $ D $ and the maximal subgroup when $ n = 1 $.

ABSTRACT

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 $F$.

Motivation & Objective

• Address the open conjecture on the non-existence of solvable maximal subgroups in $ \mathrm{GL}_n(D) $ for $ n \geq 2 $, particularly for non-abelian solvable groups.
• Generalize previous results by showing that subnormal subgroups of $ \mathrm{GL}_n(D) $ cannot contain non-abelian solvable maximal subgroups when $ n > 1 $.
• Investigate structural constraints on division rings and their subgroups when such maximal subgroups exist.
• Characterize the case $ n = 1 $ where such subgroups can exist, identifying conditions under which $ D $ is a cyclic algebra of prime degree.
• Establish that the existence of such maximal subgroups forces $ D $ to be a finite-dimensional division algebra over its center with specific Galois-theoretic structure.

Proposed method

• Use the structure of subnormal subgroups in $ \mathrm{GL}_n(D) $ and properties of maximal abelian subgroups within them.
• Apply the Cartan-Brauer-Hua theorem to show that if a maximal subgroup generates the full matrix algebra, then the group must be simple modulo its center.
• Employ the theory of locally finite and locally solvable groups to analyze the finiteness and structure of subgroups in division rings.
• Use Galois theory and group ring techniques to analyze the action of the maximal subgroup on its maximal abelian subfield, establishing isomorphisms with Galois groups.
• Apply Kaplansky’s theorem on PI-rings and Artinian rings to deduce finiteness and simplicity of the group algebra generated by the subgroup.
• Use the fundamental theorem of Galois theory to show that the quotient of the maximal subgroup modulo its center is isomorphic to a cyclic group of prime order.

Experimental results

Research questions

• RQ1 Under what conditions can a non-abelian solvable maximal subgroup exist in a subnormal subgroup of $ \mathrm{GL}_n(D) $ for $ n > 1 $?
• RQ2 What structural constraints does the existence of such a maximal subgroup impose on the division ring $ D $?
• RQ3 Can the center of $ D $ and the degree $ n $ be determined when such a maximal subgroup exists?
• RQ4 How does the Galois structure of a maximal subfield relate to the group structure of the maximal subgroup?
• RQ5 Is it possible for $ D $ to be non-cyclic or of composite degree if such a maximal subgroup exists?

Key findings

• If a subnormal subgroup $ G \leq \mathrm{GL}_n(D) $ contains a non-abelian solvable maximal subgroup, then $ n = 1 $, meaning such subgroups cannot exist in $ \mathrm{GL}_n(D) $ for $ n > 1 $.
• Under the given conditions, $ D $ must be a cyclic algebra of prime degree $ p $ over its center $ F $, with $ [D:F] = p^2 $.
• The maximal subgroup $ M $ is isomorphic to $ \mathbb{Z}_p $ modulo its center, and $ M/K^* \cap G \cong \mathbb{Z}_p $, where $ K $ is a maximal subfield of $ D $.
• The fixed field of the Galois action is $ F $, and $ K/F $ is a finite Galois extension with Galois group isomorphic to $ \mathbb{Z}_p $.
• The $ FC $-center of $ M $ is $ K^* \cap G $, which is also the Fitting subgroup of $ M $, indicating a central role in the group's structure.
• Every element $ x \in M \setminus K $ satisfies $ x^p \in F $, and $ D = F[M] = \bigoplus_{i=1}^p Kx^i $, showing that $ D $ is a crossed product algebra with a cyclic structure.