On a group-theoretical generalization of the Gauss formula

TL;DR

This paper explores a group-theoretical extension of Gauss's formula, providing new characterizations of finite cyclic groups through automorphism counts.

Contribution

It introduces a novel generalization of Gauss's formula based on automorphism functions, offering fresh insights into the structure of finite cyclic groups.

Findings

Characterizations of finite cyclic groups derived from automorphism counts

A generalized Gauss formula applicable to finite groups

New theoretical connections between automorphisms and group structure

Abstract

In this paper, we discuss a group-theoretical generalization of the well-known Gauss formula involving the functionthat counts the number of automorphisms of a finite group. This gives several characterizations of finite cyclic groups.