Strongly generating elements in finite and profinite groups

TL;DR

This paper investigates the maximum possible difference between the expected number of elements needed to generate a finite group and the expected number needed when including a specific element, exploring the generative properties of finite and profinite groups.

Contribution

It provides bounds and insights into how much adding a fixed element can reduce the expected number of generators in finite groups.

Findings

The difference $e(G)-e(G,g)$ can be arbitrarily large for certain groups.

Bounds are established for the maximum difference in various classes of groups.

The results deepen understanding of generative properties in finite and profinite groups.

Abstract

Given a finite group $G$ and an element $g\in G$, we may compare the expected number $e(G)$ of elements needed to generate $G$ and the expected number $e(G,g)$ of elements of $G$ needed to generate $G$ together with $g.$ We address the following question: how large can the difference $e(G)-e(G,g)$ be?