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 | + 1, where |G | 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 | − p + 2.

Lie Nilpotency Indices of Modular Group Algebras

“…A.Shalev in [16] began the study of the question when the Lie nilpotent group algebra KG has the maximal lower Lie nilpotency index. In [6,16] was given the complete description of such Lie nilpotent group algebras. In [4,5] we obtained the full description of the Lie nilpotent group algebras KG with upper/lower almost maximal Lie nilpotent indices.…”

“…Clearly, |γ 2 (G/H)| = 2 n−1 and t L (K[G/H]) = |γ 2 (G/H)| + 1. So by Lemma 3 of [6] and by Theorem 1 of [6] the group γ 2 (G/H) is either a cyclic 2-group or C 2 × C 2 . If γ 2 (G/H) is a cyclic 2-group, then by (a) of Lemma III.…”

“…Let t L (KG) = |G ′ |−4p+5, where p = char(K). If either n = 1 or p = n = 2, then according to Theorem 1 of [6], and to Corollary 1 of [5], we get t L (KG) ≥ |G ′ |. So in the sequel we assume that either p = 2 and n ≥ 3 or n ≥ 2.…”

“…7.1 ([10], p.300) we have that γ 2 (G) is abelian, so it is isomorphic to either C 2 n−1 × C 2 or C 2 n . If γ 2 (G) is cyclic, then by Theorem 1 of [6] we get t L (KG) = |γ 2 (G)| + 1, so we do not consider this case. On the other hand, if γ 2 (G/H) ∼ = C 2 × C 2 , then |γ 2 (G)| = 8.…”

Lie nilpotency indices of symmetric elements under oriented involutions in group algebras

“…Assume that t((F G) + ) = |G ′ | + 1. As G ′ is a finite p-group, if G ′ is not cyclic, from [4], we know that t((F G) + ) ≤ t L (F G) < |G ′ | + 1 and we get a contradiction. Thus, G ′ is cyclic.…”