Group algebras with centrally metabelian unit groups

Abstract

Given a field K of characteristic p > 2 and a finite group G, necessary and sufficient conditions for the unit group U (KG) of the group algebra KG to be centrally metabelian are obtained. It is observed that U (KG) is centrally metabelian if and only if KG is Lie centrally metabelian.

On the derived length of units in group algebra

“…Only a few results have been proved. Shalev in [20], Kurdics in [13], Sahai and Chandra in [18], [19], [9] and [10] have investigated group algebras with units having derived length at most two and three, respectively, over fields of finite characteristic. Baginski in [1] showed that if G is a finite non-abelian p-group such that G ′ is cyclic, then the derived length of U is ⌈log 2 (|G ′ | + 1)⌉, where ⌈r⌉ denotes the minimal integer not smaller than r for a real number r. This result was extended by Balogh and Li to an arbitrary group G with a cyclic derived subgroup of p-power order p > 2 in [2].…”

On group algebras with unit groups of derived length at most four

“…Shalev [7] has classified group algebras of finite groups over fields of odd characteristic whose unit group is metabelian, and Kurdics [8] has done the same for even characteristic. Further, a necessary and sufficient condition for U to be centrally metabelian, that is, (U (2) , U ) = 1, is given by Sahai [9]. Also, characterisation of group algebras over fields of odd characteristic such that (U (2) , U ′ ) = 1 has been given by Sahai [10] and the same with U satisfying (U (2) , U (2) ) = 1 has been investigated by Chandra and Sahai [11], [12].…”

On group algebras with unit groups of derived length three\\ in characteristic three

“…This work was completed by Coleman and Sandling [7] and independently by Kurdics [9] for p = 2. For p = 2, a complete description of group algebras KG with centrally metabelian unit groups is given in [13]. The group algebras with γ 3 (δ 1 (U )) = 1 have been listed in [15].…”