Units in group rings and blocks of Klein four or dihedral defect

TL;DR

This paper studies units of even order in integral group rings of finite groups, using new methods involving reductions modulo 4, to derive restrictions and answer questions about prime graphs for specific groups.

Contribution

It introduces an improved lattice method for analyzing units in group rings at p=2, especially for blocks with Klein four or dihedral defect groups, advancing understanding of units and prime graphs.

Findings

Restrictions on units of even order in group rings.

Disproof of units of order 2p in certain almost simple groups.

Affirmative answers to the Prime Graph Question for many groups.

Abstract

We obtain restrictions on units of even order in the integral group ring $\mathbb{Z}G$ of a finite group $G$ by studying their actions on the reductions modulo $4$ of lattices over the $2$-adic group ring $\mathbb{Z}_2G$. This improves the "lattice method" which considers reductions modulo primes $p$, but is of limited use for $p=2$ essentially due to the fact that $1\equiv -1 \ (\textrm{mod }2)$. Our methods yield results in cases where $\mathbb Z_2 G$ has blocks whose defect groups are Klein four groups or dihedral groups of order $8$. This allows us to disprove the existence of units of order $2p$ for almost simple groups with socle $\operatorname{PSL}(2,p^f)$ where $p^f\equiv \pm 3 \ (\textrm{mod } 8)$ and to answer the Prime Graph Question affirmatively for many such groups.