Bases of Permutation Groups and Boolean Representable Simplicial Complexes
Stuart Margolis, John Rhodes
TL;DR
This paper develops a bridge between bases of permutation groups and Boolean Representable Simplicial Complexes by associating to $(X,G)$ a lattice $L(X,G)$ with join-generators $X$, yielding a BRSC $\mathcal{B}(X,G)$. It proves that a subset $Y$ is independent in $\mathcal{B}(X,G)$ if and only if $Y$ is irredundant, and that $B$ is a base precisely when $\mathrm{Cl}(B)=X$, thereby recasting bases in BRSC language. The authors provide concrete examples, including $(P_{2}(X),G)$ and simple-group related constructions, and formulate a conjecture about simple non-abelian groups whose resolution would yield a new proof of the Feit-Thompson theorem. This work thus connects lattice-theoretic independence to core questions in computational and abstract group theory, with potential consequences for proving deep finite-group results.
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.