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Locally solvable maximal subgroups in division rings

Huynh Viet Khanh, Bui Xuan Hai

Abstract

Let $D$ be a division ring with center $F$, and $G$ an almost subnormal subgroup of $D^*$. In this paper, we show that if $G$ contains a non-abelian locally solvable maximal subgroup, then $D$ must be a cyclic algebra of prime degree over $F$. Moreover, it is proved that every locally nilpotent maximal subgroup of $G$ is abelian.

Locally solvable maximal subgroups in division rings

Abstract

Let be a division ring with center , and an almost subnormal subgroup of . In this paper, we show that if contains a non-abelian locally solvable maximal subgroup, then must be a cyclic algebra of prime degree over . Moreover, it is proved that every locally nilpotent maximal subgroup of is abelian.

Paper Structure

This paper contains 3 sections, 17 theorems, 15 equations.

Table of Contents

  1. Introduction and statement of results
  2. Examples of maximal subgroups
  3. Locally solvable maximal subgroups

Key Result

Theorem 1.1

Let $D$ be a division ring with center $F$, and $G$ an almost subnormal subgroup of $D^*$. If $M$ is a locally nilpotent maximal subgroup of $G$, then $M$ is abelian.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Example 1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • ...and 9 more