Maximal subgroups of SL(n, Z)

TL;DR

This paper investigates the structure of maximal subgroups in SL(n,Z), demonstrating their diverse types, existence of many such subgroups, and providing specific examples with unique properties.

Contribution

The paper establishes the existence of various types of maximal subgroups in SL(n,Z), including those with non-dense orbits and specific permutation representations, expanding understanding of subgroup diversity.

Findings

Existence of continuously many maximal subgroups

Construction of a maximal subgroup with no dense orbits on projective space

A faithful primitive permutation representation of PSL(n,Z) that is not 2-transitive

Abstract

We establish the existence of maximal subgroups of various diferent natures in SL(n,Z). In particular, we prove that there are continuously many maximal subgroups, we provide a maximal subgroup whose action on the projective space has no dense orbits, and we produce a faithful primitive permutation representation of PSL(n,Z) which is not 2-transitive.