Towards the Automorphism Conjecture I: Combinatorial Control

TL;DR

This paper advances the understanding of automorphism counts in ordered sets by linking group classification to combinatorial orbit analysis, aiming to prove an exponential bound in the Automorphism Conjecture.

Contribution

It introduces a novel approach connecting simple group classification with orbit adjacency analysis to bound automorphisms in ordered sets.

Findings

Identifies structures that prevent exponential automorphism bounds.

Provides conditions under which automorphism counts can be exponentially bounded.

Highlights structures that inflate automorphism numbers beyond bounds.

Abstract

This paper exploits adjacencies between the orbits of an ordered set P and a consequence of the classification of finite simple groups to, in many cases, exponentially bound the number of automorphisms. Results clearly identify the structures which currently prevent the proof of such an exponential bound, or which indeed inflate the number of automorphisms beyond such a bound. This is a first step towards a possible resolution of the Automorphism Conjecture for ordered sets.