# Orders of units in integral group rings and blocks of defect $1$

## TL;DR

This paper investigates the structure of units in integral group rings of finite groups with prime order Sylow subgroups, establishing conditions for units of composite order and applying results to solve the Prime Graph Question for various simple groups.

## Contribution

It provides a new criterion linking units of order pq in integral group rings to elements of the same order in the group, advancing understanding of the Prime Graph Question.

## Key findings

- Units of order pq exist iff the group contains an element of order pq
- Solved the Prime Graph Question for most sporadic simple groups
- Extended results to some simple groups of Lie type

## Abstract

We show that if the Sylow $p$-subgroup of a finite group $G$ is of order $p$, then the normalized unit group of the integral group ring of $G$ contains a normalized unit of order $pq$ if and only if $G$ contains an element of order $pq$, where $q$ is any prime. We use this result to answer the Prime Graph Question for most sporadic simple groups and some simple groups of Lie type, including a new infinite series of such groups. Our methods are based on the understanding of blocks of cyclic defect and Young tableaux combinatorics.