Primitive groups, road closures, and idempotent generation

TL;DR

This paper investigates conditions under which certain semigroups generated by permutation groups and non-permutations are idempotent-generated, extending previous results and introducing new properties like the road closure condition.

Contribution

The authors prove that for large enough rank and degree, the permutation group must be symmetric or alternating if the semigroup is idempotent-generated, and introduce the road closure condition for the case k=2.

Findings

For k ≥ 6 and n ≥ 2k+1, G is symmetric or alternating.

Idempotent generation for all maps of fixed rank k implies strong group properties.

The road closure condition is a new criterion stronger than primitivity.

Abstract

We are interested in semigroups of the form $\langle G,a\rangle\setminus G$, where $G$ is a permutation group of degree $n$ and $a$ a non-permutation on the domain of $G$. A theorem of the first author, Mitchell and Schneider shows that, if this semigroup is idempotent-generated for all possible choices of $a$, then $G$ is the symmetric or alternating group of degree $n$, with three exceptions (having $n=5$ or $n=6$). Our purpose here is to prove stronger results where we assume that $\langle G,a\rangle\setminus G$ is idempotent-generated for all maps of fixed rank $k$. For $k\ge6$ and $n\ge2k+1$, we reach the same conclusion, that $G$ is symmetric or alternating. These results are proved using a stronger version of the \emph{$k$-universal transversal property} previously considered by the authors. In the case $k=2$, we show that idempotent generation of the semigroup for all choices of $a$ is equivalent to a condition on the permutation group $G$, stronger than primitivity, which we call the \emph{road closure condition}. We cannot determine all the primitive groups with this property, but we give a conjecture about their classification, and a body of evidence (both theoretical and computational) in support of the conjecture. The paper ends with some problems.