Lie Properties of the Group Algebra and the Nilpotency Class of the Group of Units

Abstract

We describe the upper and lower Lie nilpotency index of a modular group algebra ‫ކ‬G of some metabelian group G and apply these results to determine the nilpotency class of the group of units, extending certain results of Shalev without restriction to finite groups. A characterization of modular group algebras ‫ކ‬G with group of units of class 3 is given, which was obtained by Rao and Sandling for finite groups G. ᮊ

Modular Group Algebras with Almost Maximal Lie Nilpotency Indices

“…Assuming that KG has upper almost maximal Lie nilpotency index, the statement is a consequence of the previous lemmas. The converse is obvious by (3).…”

“…Now, let us compute the weak complement of γ 3 (G ) in γ 2 (G ) (see [3], p. 34). It is easy to see that in the notation of [3] ν(a 2 b) = 2 in both cases and the weak complement is A = a . As in the case 1.b we have t L (KG ) = 7 = |G |, a contradiction.…”

“…It follows that d(3 i +1) = 1 for 0 ≤ i ≤ n − 2, d ( j) = 0 for j = 3 i + 1 and j > 1, where 3 d(k) = [D (k) (G/H) : D (k+1) (G/H)].Clearly, |γ 2 (G/H)| = 3 n−1 and t L (K[G/H]) = |γ 2 (G/H)| + 1. By Lemma 3 of [4]…”

“…But the question of when t L (KG) = t L (KG) for char (K) = 2, 3 is still open in general. Several important results on this topic were obtained in [3]. We say that a Lie nilpotent group algebra KG has upper almost maximal Lie nilpotency index, if t L (KG) = |G | − p + 2.…”

Units in finite dihedral and quaternion group algebras

“…The study of the group of units is one of the classical topics in group ring theory. Results obtained in this direction are useful for the investigation of Lie properties of group rings, isomorphism problem and other open questions in this area [1]. In [2], Bovdi gave a comprehensive survey of results concerning the group of units of a modular group algebra of characteristic p. There is a long tradition on the study of the unit group of finite group algebras [3][4][5][6][7][8][9][10][11][12].…”