New-type Quasirandom Groups and Applications

TL;DR

This paper introduces a generalized concept of quasirandom groups, extending key results to broader classes, and explores their structural properties and product set growth, with applications to groups of rigid motions over finite fields.

Contribution

It generalizes the definition of quasirandom groups and extends known results, including conditions for element tuples and exponential product set growth, with specific insights into rigid motion groups.

Findings

Provided conditions for tuples satisfying product equations.

Established criteria for exponential growth of product sets.

Described structures of rigid motion groups over finite fields.

Abstract

This paper aims to introduce a more general definition of quasirandom groups and generalize several well-known results in the literature in this new setting. More precisely, let $G$ be a semi-direct product of groups and $X\subseteq G$, we provide conditions such that one can find tuples $(x_0, \ldots, x_k)\in X^{k+1}$ satisfying $x_1x_2\ldots x_k=x_0$ or conditions to guarantee that the product set $XX$ grows exponentially. In a special case of the group of rigid-motions in the plane over an arbitrary finite field, our results offer a reasonably complete description of structures of this group.