Some group theory changes for the OSCAR book (#3242)

* Some group theory changes for the OSCAR book

* nicer grunwald_wang interface

* more

experimental/GModule/Cohomology.jl

@@ -8,6 +8,8 @@ import Oscar: GAPWrap, pc_group, fp_group, direct_product, direct_sum
 import AbstractAlgebra: Group, Module
 import Base: parent
 
+import Oscar: pretty, Lowercase, @show_name, @show_special
+
 function __init__()
   Hecke.add_verbose_scope(:GroupCohomology)
   Hecke.add_assert_scope(:GroupCohomology)
@@ -104,11 +106,12 @@ Base.hash(a::MultGrpElem, u::UInt = UInt(1235)) = hash(a.data. u)
 end
 
 function Base.show(io::IO, C::GModule)
-  AbstractAlgebra.@show_name(io, C)
-  AbstractAlgebra.@show_special(io, C)
+  @show_name(io, C)
+  @show_special(io, C)
 
+  io = pretty(io)
   io = IOContext(io, :compact => true)
-  print(io, "G-module for ", C.G, " acting on ", C.M)# , "\nvia: ", C.ac)
+  print(io, "G-module for ", Lowercase(), C.G, " acting on ", Lowercase(), C.M)# , "\nvia: ", C.ac)
 end
 
 """

experimental/GModule/GModule.jl

@@ -3,6 +3,18 @@ using Oscar
 using Hecke
 import Hecke: data
 
+#XXX: clash of names!
+#   gmodule(k, C) vs gmodule_ver(k, C)
+#   first does a "restriction of scalars" or blow up with the rep mat
+#   second tries to conjugate down to k
+
+import Oscar: gmodule, GAPWrap
+import Oscar.GrpCoh: MultGrp, MultGrpElem
+
+import AbstractAlgebra: Group, Module
+import Base: parent
+
+
 #XXX: have a type for an implicit field - in Hecke?
 #     add all(?) the other functions to it
 function relative_field(m::Map{<:AbstractAlgebra.Field, <:AbstractAlgebra.Field})
@@ -59,10 +71,10 @@ julia> C = gmodule(CyclotomicField, C);
 julia> h = subfields(base_ring(C), degree = 2)[1][2];
 
 julia> restriction_of_scalars(C, h)
-G-module for G acting on Vector space of dimension 4 over number field of degree 2 over QQ
+G-module for G acting on vector space of dimension 4 over number field of degree 2 over QQ
 
 julia> restriction_of_scalars(C, QQ)
-G-module for G acting on Vector space of dimension 8 over rational field
+G-module for G acting on vector space of dimension 8 over rational field
 
 ```
 """
@@ -200,17 +212,6 @@ function invariant_lattice_classes(M::GModule, phi::Map)
 end
 
 
-#XXX: clash of names!
-#   gmodule(k, C) vs gmodule_ver(k, C)
-#   first does a "restriction of scalars" or blow up with the rep mat
-#   second tries to conjugate down to k
-
-import Oscar:gmodule, GAPWrap
-import Oscar.GrpCoh: MultGrp, MultGrpElem
-
-import AbstractAlgebra: Group, Module
-import Base: parent
-
 function __init__()
   add_verbose_scope(:BruecknerSQ)
   set_verbose_level(:BruecknerSQ, 0)
@@ -262,8 +263,12 @@ function irreducible_modules(G::Oscar.GAPGroup)
   return IM
 end
 
-"""
-  G acting on M trivially, ie. g(m) == m.
+@doc raw"""
+    trivial_gmodule(G::Group, M::GrpAbFinGen)
+    trivial_gmodule(G::Group, M::AbstractAlgebra.FPModule)
+
+Return the `G`-module over the underlying set `M` with trivial `G`-action,
+i.e., `g(m) == m` for all $g\in G$ and $m\in M$.
 """
 function trivial_gmodule(G::Oscar.GAPGroup, M::Union{GrpAbFinGen, AbstractAlgebra.FPModule})
   I = hom(M, M, gens(M))
@@ -351,6 +356,11 @@ function irreducible_modules(::ZZRing, G::Oscar.GAPGroup)
   return [gmodule(ZZ, m) for m in z]
 end
 
+"""
+    gmodule(k::Field, C::GModule)
+
+TODO
+"""
 function gmodule(::typeof(CyclotomicField), C::GModule)
   @assert isa(base_ring(C), QQAbField)
   d = dim(C)
@@ -382,18 +392,16 @@ function gmodule(k::Nemo.fpField, mC::Hecke.MapClassGrp)
   return gmodule(k, gmodule(ray_class_field(mC)))
 end
 
-
-import Base: ^
-function ^(C::GModule{<:Any, <:AbstractAlgebra.FPModule{nf_elem}}, phi::Map{AnticNumberField, AnticNumberField})
+function Base.:^(C::GModule{<:Any, <:AbstractAlgebra.FPModule{nf_elem}}, phi::Map{AnticNumberField, AnticNumberField})
   F = free_module(codomain(phi), dim(C))
   return GModule(group(C), [hom(F, F, map_entries(phi, matrix(x))) for x = C.ac])
 end
 
-function ^(C::GModule{<:Any, T}, h::Map{S, S}) where T <: S where S
+function Base.:^(C::GModule{<:Any, T}, h::Map{S, S}) where T <: S where S
   return GModule(group(C), [inv(h)*x*h for x = C.ac])
 end
 
-function ^(C::GModule{<:Any, <:AbstractAlgebra.FPModule{QQAbElem}}, phi::Map{QQAbField, QQAbField})
+function Base.:^(C::GModule{<:Any, <:AbstractAlgebra.FPModule{QQAbElem}}, phi::Map{QQAbField, QQAbField})
   F = free_module(codomain(phi), dim(C))
   return GModule(F, group(C), [hom(F, F, map_entries(phi, matrix(x))) for x = C.ac])
 end
@@ -523,6 +531,12 @@ function Hecke.frobenius(K::FinField, i::Int=1)
   MapFromFunc(K, K, x->Hecke.frobenius(x, i), y -> Hecke.frobenius(x, degree(K)-i))
 end
 
+@doc raw"""
+    gmodule_minimal_field(C::GModule)
+    gmodule_minimal_field(k::Field, C::GModule)
+
+Return TODO
+"""
 function gmodule_minimal_field(C::GModule{<:Any, <:AbstractAlgebra.FPModule{fpFieldElem}})
   return C
 end
@@ -545,6 +559,9 @@ function gmodule_minimal_field(C::GModule{<:Any, <:AbstractAlgebra.FPModule{nf_e
   return _minimize(C) 
 end
 
+"""
+    gmodule_over(Field)
+"""
 function gmodule_over(k::FinField, C::GModule{<:Any, <:AbstractAlgebra.FPModule{<:FinFieldElem}}; do_error::Bool = false)
   #mathematically, k needs to contain the character field
   #only works for irreducible modules
@@ -1515,6 +1532,8 @@ export is_decomposable
 export is_G_hom
 export restriction_of_scalars
 export trivial_gmodule
+export gmodule_minimal_field
+export gmodule_over
 
 ## Fill in some stubs for Hecke
 
@@ -1568,4 +1587,5 @@ export is_decomposable
 export is_G_hom
 export restriction_of_scalars
 export trivial_gmodule
-
+export gmodule_minimal_field
+export gmodule_over

experimental/GModule/GaloisCohomology.jl

@@ -1,5 +1,6 @@
 module GaloisCohomology_Mod
 using Oscar
+import Oscar: gmodule
 import Oscar: GrpCoh
 import Oscar.GrpCoh: CoChain, MultGrpElem, MultGrp, GModule, is_consistent, 
                      Group
@@ -1609,7 +1610,7 @@ function relative_brauer_group(K::AnticNumberField, k::Union{QQField, AnticNumbe
   return B, B.map
 end
 
-function (B::RelativeBrauerGroup)(M::GModule{PermGroup, AbstractAlgebra.Generic.FreeModule{nf_elem}})
+function (B::RelativeBrauerGroup)(M::GModule{<:Group, AbstractAlgebra.Generic.FreeModule{nf_elem}})
   @assert B.K == base_ring(M)
   c = factor_set(M, B.mG)
   return preimage(B.map, c)
@@ -1664,6 +1665,17 @@ function Hecke.AlgAss(a::RelativeBrauerGroupElem)
   return A = AlgAss(parent(a).map(a), parent(a).mG, parent(a).mkK)
 end
 
+function Hecke.grunwald_wang(b::RelativeBrauerGroupElem)
+  d1 = Dict((x=>Int(order(y))) for (x,y) = b.data if isa(x, NumFieldOrdIdl))
+  d2 = Dict((x=>Int(order(y))) for (x,y) = b.data if !isa(x, NumFieldOrdIdl))
+
+  if length(d2) == 0
+    return Hecke.grunwald_wang(d1)
+  else
+    return Hecke.grunwald_wang(d1, d2)
+  end
+end
+
 function local_index(C::GModule{<:Oscar.GAPGroup, Generic.FreeModule{nf_elem}}, p::Union{Integer, ZZRingElem})
   K = base_ring(C)
   k, mkK = Oscar.GModuleFromGap._character_field(C)