Cohomology

Dual(G): GrpAb -> GrpAb, Map

Computes the dual group \(G^*\) of \(G\) and a map \(M\) from \(G\times G^* \to {\mathbb{Z}}/m{\mathbb{Z}}\) for \(m\) the exponent of \(G\) that allows \(G^*\) to act on \(G\). The group \(G\) must be finite.

H2_G_QmodZ(G): GrpAb -> GrpAb, Map

Computes \(H := H^2(G, {\mathbb{Q}}/{\mathbb{Z}})\) and a map \(f : H \to (G\times G \to {\mathbb{Z}}/m{\mathbb{Z}})\) that will give the cocycles as maps from \(G\times G \to {\mathbb{Z}}/m{\mathbb{Z}}\), \(m := \#G\).

Res_H2_G_QmodZ(U, H2): GrpAb, GrpAb -> GrpAb, Map

For a subgroup \(U\) of \(G\) and \(H2 = H^2(G, {\mathbb{Q}}/{\mathbb{Z}})\) computes \(H^2(U, {\mathbb{Q}}/{\mathbb{Z}})\) in a compatible way together with the restriction map into \(H2\).

The abelian group \(H2\) must be the result of H2_G_QmodZ as this function relies on the attributes stored in there.