Modular Group Algebras with Almost Maximal Lie Nilpotency Indices. I

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 (or lower) Lie nilpotency index is at most |G|+ 1, where |G| is the order of the commutator subgroup. The authors have previously determined the groups G for which this index is maximal and here they determine the G for which it is ‘almost maximal’, that is the next highest possible value, namely |G| − p+ 2. Let R be an associative algebra with identity. The algebra R can be regarded as a Lie algebra, called the associated Lie algebra of R, via the Lie commutator [x, y] = xy − yx, for every x, y ∈ R. Set [x1, . . . , xn] = [[x1, . . . , xn−1], xn], where x1, . . . , xn ∈ R. The n-th lower Lie power R [n] of R is the associative ideal generated by all the Lie commutators [x1, . . . , xn], where R[1] = R and x1, . . . , xn ∈ R. By induction, we define the n-th upper Lie power R(n) of R as the associative ideal generated by all the Lie commutators [x, y], where R(1) = R and x ∈ R(n−1), y ∈ R. The algebra R is called Lie nilpotent (respectively upper Lie nilpotent) if there exists m such that R[m] = 0 (R(m) = 0). The algebra R is called Lie hypercentral if for each sequence {ai} of elements of R there exists some n such that [a1, . . . , an] = 0. The minimal integers m,n such that R[m] = 0 and R(n) = 0 are called the Lie nilpotency index and the Mathematics Subject Classification: 16S34, 17B30.