Determining sets and determining numbers of finite groups

TL;DR

This paper investigates the properties of determining sets and numbers in finite groups, classifies groups with minimal determining numbers, and explores the relationship between determining and generating numbers, introducing DEG-groups.

Contribution

It classifies finite groups with small determining numbers, proves that finite simple and nilpotent groups are DEG-groups, and constructs groups with specific relationships between determining and generating numbers.

Findings

Finite groups with determining number 0 or 1 are classified.

Finite simple and nilpotent groups are DEG-groups.

Existence of groups with prescribed relationships between lpha(G) and gamma(G).

Abstract

Let $G$ be a group. A subset $D$ of $G$ is a determining set of $G$, if every automorphism of $G$ is uniquely determined by its action on $D$. The determining number of $G$, denoted by $\alpha(G)$, is the cardinality of a smallest determining set. A generating set of $G$ is a subset such that every element of $G$ can be expressed as the combination, under the group operation, of finitely many elements of the subset and their inverses. The cardinality of a smallest generating set of $G$, denoted by $\gamma(G)$, is called the generating number of $G$. A group $G$ is called a DEG-group if $\alpha(G)=\gamma(G)$. The main results of this article are as follows. Finite groups with determining number $0$ or $1$ are classified; Finite simple groups and finite nilpotent groups are proved to be DEG-groups; A finite group is a normal subgroup of a DEG-group and there is an injective mapping from the set all finite groups to the set of finite DEG-groups; Nilpotent groups of order $n$ which have the maximum determining number are classified; For any integer $k\geq 2$, there exists a group $G$ such that $\alpha(G)=2$ and $\gamma(G)\geq k$.