Unit groups of group algebras of some small groups

Let FG be a group algebra of a group G over a field F and U (FG) the unit group of FG. It is a classical question to determine the structure of the unit group of the group algebra of a finite group over a finite field. In this article, the structure of the unit group of the group algebra of the non-abelian group G with order 21 over any finite field of characteristic 3 is established. We also characterize the structure of the unit group of FA4 over any finite field of characteristic 3 and the structure of the unit group of FQ12 over any finite field of characteristic 2, where Q12 = 〈x, y; x6 = 1, y2 = x3, xy = x−1〉.