Modular Group Algebras with Almost Maximal Lie Nilpotency Indices

Abstract

Let \(K\) be a field of positive characteristic \(p\) and \(KG\) the group algebra of a group \(G\). It is known that, if \(KG\) is Lie nilpotent, then its upper (and lower) Lie nilpotency index is at most \(|G^{\, \prime}|+1\), where \(|G^{\, \prime}|\) is the order of the commutator subgroup. The authors previously determined those groups \(G\) for which this index is maximal and here they determine the groups \(G\) for which it is `almost maximal', that is, it takes the next highest possible value, namely \(|G^{\, \prime}|-p+2\).