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Explicit \(K_1\) of some modular group rings. (English) Zbl 1093.19001

The author surveys methods to compute the \(K_1\) of the group ring of a finite group over a finite prime field. The emphasis is on the case where the prime field has two elements. The methods are illustrated on examples, like dihedral groups and the symmetric group on four letters. The answers are very explicit; they do not only give the group structure of \(K_1\), but also explicit units in the group ring corresponding to generators of \(K_1\).

MSC:

19B28 \(K_1\) of group rings and orders
19B14 Stability for linear groups
19A31 \(K_0\) of group rings and orders
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16U60 Units, groups of units (associative rings and algebras)
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