Tuple regularity and $k$-ultrahomogeneity for finite groups

TL;DR

This paper introduces and classifies new notions of $k$-ultrahomogeneity and $ au$-tuple regularity in finite groups, extending concepts from graph theory and automorphism transitivity, with implications for group theory and algorithms.

Contribution

It defines the concepts of $k$-ultrahomogeneity and $ au$-tuple regularity for finite groups and provides a classification for groups satisfying these properties.

Findings

Every 2-tuple regular finite group is ultrahomogeneous.

Classification of $k$-ultrahomogeneous and $ au$-tuple regular finite groups for $k, au \,\geq 2$.

Connections established between group properties and algorithmic interpretations.

Abstract

For $k, \ell \in \mathbb{N}$, we introduce the concepts of $k$-ultrahomogeneity and $\ell$-tuple regularity for finite groups. Inspired by analogous concepts in graph theory, these form a natural generalization of homogeneity, which was studied by Cherlin and Felgner and Li as well as automorphism transitivity, which was investigated by Zhang. Additionally, these groups have an interesting algorithmic interpretation. We classify the $k$-ultrahomogeneous and $\ell$-tuple regular finite groups for $k, \ell \geq 2$. In particular, we show that every 2-tuple regular finite group is ultrahomogeneous.