On the widths of finite groups (I)

TL;DR

This paper investigates the properties of finite groups by examining the widths of their prime divisors and element orders, providing new results and a survey on these measures.

Contribution

It introduces and analyzes the concepts of order width and spectrum width of finite groups, offering new theoretical results and a comprehensive survey.

Findings

Characterization of groups with small order width

Results on groups with limited spectrum width

Survey of existing knowledge on group widths

Abstract

Let $G$ be a finite group, $\pi(G)$ be the set of prime divisors dividing the order of $G$ and $\pi_e(G)$ (spectrum) denote the set of element orders of $G$. We define $w_o(G)$ = $|\pi(G)|$ the width of order of $G$ and $w_s(G)$ = max$\{|{\pi}(k)| | k \in \pi_e(G)\}$ the width of spectrum of $G$. In this paper, we discuss the cases of $w_o(G)$ and $w_s(G)$ are small, prove several new results and give a survey about the two widths of groups.