TITLE: Units in a group algebra
QUESTION [6 upvotes]: Let k be a field and let G be a finite group.  I would like to know if there is any nice description of the group of units in the group algebra kG. (If there is no nice answer in this generality, assume that the characteristic of k is p and that G is a p-group.)

REPLY [12 votes]: If $k$ has characteristic $p$ and $G$ is a finite $p$-group, then the augmentation ideal $J$ of $kG$ is nilpotent and of codimension one, hence coincides with the Jacobson radical. This means that the units of $kG$ are precisely the complement $kG \setminus J = \{ \sum_g c_g g \mid \sum_g c_g \neq 0 \}$ of the augmentation ideal. 
Is this the description you asked for, or do you rather want a procedure for determining the isomorphy type of the group of units (which is the product of $k^{\times}$ and a $p$-group of order $|k|^{|G|-1}$) from that of $G$? When $G$ is abelian, this problem is completely solved in R. Sandling's paper Units in the modular group algebra of a ﬁnite abelian $p$-group (J. Pure Appl. Algebra 1984). In the non-abelian case there are many papers in the literature devoted to particular $p$-groups $G$, but as noted in Stefan Kohl's answer, the problem is not understood in full generality.

REPLY [11 votes]: If $k$ is a field of characteristic $p>0$ and $G$ is a finite $p$-group then it is well-known that the Jacobson radical of $kG$ coincides with the augmentation ideal $\omega(kG)$ of $kG$. The group of units of $kG$ is given by $kG\backslash \omega(kG)$ and is isomorphic to the direct product $k^\times \times (1+\omega(kG))$, where $k^\times=k\backslash\{0\}$. Some other results on the group of units of group algebras can be found, e.g., in the books "S.K. Sehgal: Topics in group rings", "D. Passman: The algebraic structure of group rings", "I. Passi: group rings and their augmentation ideal", and in many papers of the existing literature on the subject.