Unit group and first K-group

unit_group(R::Union{FiniteRing, Oscar.Hecke.AbstractAssociativeAlgebra}) -> FPGroup, Map

Return a finitely presented group $G$ and a map $f \colon G \to R$, which induces an isomorphism $G \to R^\times$ is an isomorphism.

Examples

julia> R, = finite_ring(GF(2)[symmetric_group(4)]); # this the modular group ring F_2[S_4]

julia> U, mU = unit_group(R)
(Finitely presented group of order 3145728, Map: U -> finite ring)

Oscar.k1(R::Union{FiniteRing, Oscar.Hecke.AbstractAssociativeAlgebra}) -> FinGenAbGrp, Map

Return an abelian group $A$ and a map $f \colon A \to R$, such that the composition $f \to R^\times \to K_1(R)$ is an isomorphism.

Examples

julia> R, = finite_ring(GF(2)[small_group(4, 2)]);

julia> Oscar.k1(R)
((Z/2)^3, Map: (Z/2)^3 -> finite ring)