Unit groups of commutative modular group algebras (Q6601874)

The paper is rather a review of known and published results on the unit groups of the commutative modular group algebras. The authors introduce more visible and concise formulations of a number of these results. The authors also analyze some significantly inexact and unproved results (Subsection 4.4). Herewith, the authors either make corrections themselves or point to such corrections given in other papers. Some important open problems on this topic are presented in Section 5.\N\NIf \(U(RG)\) is the group of units of a group ring \(RG\), \(x \in U(RG)\) and \(x =\sum_{g\in G} x_g g\), where \(x_g\in R\), then \(x\) is called a normalized unit of \(RG\), if \(\sum_{g\in G} x_g = 1\). The normalized units form a subgroup \(V(RG)\) of \(U(RG)\). An abelian \(p\)-group \(G\) is simply presented if it has a presentation \(G = \langle X \, | \, R\rangle\), where every relation in \(R\) has the form \(px = 0\) or \(px = y\). \N\NThe authors consider the description of the unit groups of the commutative modular group algebras in the countable case (Subsection 4.1) and description of the \(p\)-component of the normalized unit groups (Subsection 4.3). Conditions for the normalized unit groups of the commutative modular group algebra to be simply presented were considered in Subsection 4.2. The authors also consider the Warfield invariants of the unit groups of group algebras (Subsection 4.5).