The probability that the product of two elements of finite group algebra is zero

TL;DR

This paper derives formulas for the probability that the product of two elements in finite group algebras equals zero, focusing on specific groups like cyclic, quaternion, and symmetric groups.

Contribution

It provides explicit formulas for zero-product probabilities in finite group algebras for certain groups, extending understanding of algebraic zero-divisors.

Findings

Formulas for cyclic groups $C_n$

Formulas for quaternion group $Q_8$

Formulas for symmetric group $S_3$

Abstract

Let $\mathbb{F}_qG$ be a finite group algebra. We denote by $P(\mathbb{F}_qG)$ the probability that the product of two elements of $\mathbb{F}_qG$ be zero. In this paper, the general formula for computing the $P(\mathbb{F}_qG)$ are established for the cyclic groups $C_n$, the Quaternion group $Q_8$ and the symmetric group $S_3$, for some cases.