# A polynomial bound for the number of maximal systems of imprimitivity of a finite transitive permutation group

## TL;DR

This paper establishes an upper bound on the number of maximal systems of imprimitivity in finite transitive permutation groups, showing it grows at most proportionally to n^{3/2}, with tighter bounds for soluble groups.

## Contribution

It provides a polynomial bound on the number of maximal systems of imprimitivity, generalizing previous results and improving understanding of subgroup structures in finite groups.

## Key findings

- Maximal systems of imprimitivity are bounded by a constant times n^{3/2}.
- For soluble groups, the bound is linear in the index, at most |G:H|-1.
- The results unify bounds for all finite groups and soluble groups.

## Abstract

We show that, there exists a constant $a$ such that, for every subgroup $H$ of a finite group $G$, the number of maximal subgroups of $G$ containing $H$ is bounded above by $a|G:H|^{3/2}$. In particular, a transitive permutation group of degree $n$ has at most $an^{3/2}$ maximal systems of imprimitivity. When $G$ is soluble, generalizing a classic result of Tim Wall, we prove a much stroger bound, that is, the number of maximal subgroups of $G$ containing $H$ is at most $|G:H|-1$.