Groups of permutations that are even on maximal proper subsets, and related monoids

TL;DR

This paper characterizes specific permutation groups and monoids based on even restrictions on maximal proper subsets, providing their structure, sizes, ranks, and minimal generators.

Contribution

It offers new descriptions, cardinalities, ranks, and minimal generating sets for the groups and monoids related to even restrictions on permutations.

Findings

Descriptions of $ ext{Gamma}_n$, $ ext{Delta}_n$, and $ ext{Sigma}_n$

Determined cardinalities and ranks of these structures

Provided minimal generating sets for each

Abstract

Let $n$ be a positive integer and let $[n]=\{1,2,\ldots,n\}$. Let $\Gamma_n$ denote the group of permutations on $[n]$ whose restrictions to maximal proper subsets of $[n]$ are even, let $\Sigma_n$ denote the monoid of transformations on $[n]$ whose injective restrictions to maximal proper subsets of $[n]$ are even and let $\Delta_n$ denote the submonoid of $\Sigma_n$ generated by transformations of rank at least $n-1$. In this paper, we present descriptions of $\Gamma_n$, $\Delta_n$ and $\Sigma_n$, determine their cardinalities and ranks, and provide minimal generating sets for each of them.