Normality degrees of finite groups

TL;DR

This paper introduces the concept of normality degree in finite groups, measuring the likelihood that a random subgroup is normal, and provides explicit formulas and limit calculations for various classes.

Contribution

It defines the normality degree of finite groups and derives explicit formulas and limit values for specific group classes, advancing understanding of subgroup normality probabilities.

Findings

Explicit formulas for normality degrees in certain finite groups

Limit values of normality degrees for various classes

Enhanced understanding of subgroup normality probabilities

Abstract

In this paper we introduce and study the concept of normality degree of a finite group $G$. This quantity measures the probability of a random subgroup of $G$ to be normal. Explicit formulas are obtained for some particular classes of finite groups. Several limits of normality degrees are also computed.