Bases of Permutation Groups and Boolean Representable Simplicial Complexes

TL;DR

This paper introduces a new combinatorial structure called Boolean representable simplicial complexes linked to permutation groups, providing insights into their bases and irredundant sequences, with implications for group theory and the Feit-Thompson Theorem.

Contribution

It defines Boolean representable simplicial complexes for permutation groups and explores their properties, offering a novel perspective that connects combinatorics and group theory.

Findings

Definition of Boolean representable simplicial complex B(X,G)

Characterization of bases via closures in B(X,G)

A conjecture related to the Feit-Thompson Theorem

Abstract

A base of a permutation group (X,G) is a subset B of X such that its pointwise stabilizer is the trivial group. A list (x1,x2, ... ,xk) of elements of X is irredundant if each element is not in the pointwise stabilizer of its predecessors. We define a Boolean representable simplicial complex B(X,G) such that a subset Y of X is independent if and only if some enumeration of its elements is irredundant. In addition Y is a base if and only if its closure is X. We give a number of examples and close with a conjecture whose solution leads to a new proof of the Feit-Thompson Theorem.