# On vanishing criteria that control finite group structure II

## TL;DR

This paper extends previous work by showing that analyzing vanishing elements of prime power order suffices for certain arithmetical properties of finite groups, further weakening the conditions needed for such results.

## Contribution

It introduces weaker criteria focusing on prime power order vanishing elements to determine finite group structure, building on prior results involving conjugacy class sizes.

## Key findings

- Vanishing elements of prime power order are sufficient for certain arithmetical group properties.
- Weaker conditions than previous criteria are established for finite group analysis.
- The results generalize earlier theorems on conjugacy class sizes and group structure.

## Abstract

In a paper by the first author it was shown that for certain arithmetical results on conjugacy class sizes it is enough to only consider the vanishing conjugacy class sizes. In this paper we further weaken the conditions to consider only vanishing elements of prime power order.