Add encode and pc_group for encoding finite pc groups into integers and back

Project.toml

@@ -35,7 +35,7 @@ AbstractAlgebra = "0.47.4"
 AlgebraicSolving = "0.10.0"
 Compat = "4.13.0"
 Distributed = "1.6"
-GAP = "0.16.0"
+GAP = "0.16.1" 
 Hecke = "0.39.1"
 JSON = "1.0.1"
 JSON3 = "1.13.2"

src/Groups/pcgroup.jl

@@ -923,3 +923,56 @@ function collector(::Type{T}, G::PcGroup) where T <: IntegerUnion
 end
 
 collector(G::PcGroup) = collector(ZZRingElem, G)
+
+# GAP wrappers for group encoding / decoding
+
+"""
+   encode(G::PcGroup)
+
+Return a `ZZRingElem` representing the polycyclic group `G`,
+using the same encoding as GAP's `CodePcGroup` and Magma's `SmallGroupEncoding`.
+Currently only defined for `PcGroup`, not `SubPcGroup`.
+
+# Examples
+```jldoctest
+julia> G = small_group(12, 2)
+Pc group of order 12
+
+julia> code = encode(G)
+266
+
+julia> H = pc_group(order(G), code)
+Pc group of order 12
+
+julia> encode(G) == encode(H)
+true
+```
+"""
+function encode(G::PcGroup)
+  return ZZ(GAP.Globals.CodePcGroup(GapObj(G))::GapInt)
+end
+
+"""
+   pc_group(order::IntegerUnion, code::IntegerUnion)
+
+Given an integer `order` and an integer `code`, return the polycyclic group it encodes.
+The accepted codes and resulting groups match those of GAP's `PcGroupCode` and Magma's `SmallGroupDecoding`.
+
+# Examples
+```jldoctest
+julia> G = small_group(12, 2)
+Pc group of order 12
+
+julia> code = encode(G)
+266
+
+julia> H = pc_group(order(G), code)
+Pc group of order 12
+
+julia> encode(G) == encode(H)
+true
+```
+"""
+function pc_group(order::IntegerUnion, code::IntegerUnion)
+  return PcGroup(GAP.Globals.PcGroupCode(GapInt(code), GapInt(order)))
+end

src/exports.jl

@@ -599,6 +599,7 @@ export elliptic_parameter
 export elliptic_surface
 export embedding
 export embedding_orthogonal_group
+export encode
 export enriques_surface_automorphism_group
 export enumerate_classes_of_lattices_with_isometry
 export epimorphism_from_free_group

test/Groups/pcgroup.jl

@@ -150,3 +150,20 @@ end
     @test is_bijective(f)
   end
 end
+
+@testset "pcgroup code and reconstruction" begin
+  groups = [
+      cyclic_group(6),
+      cyclic_group(12),
+      dihedral_group(10),
+      small_group(PcGroup, 12, 2)
+  ]
+
+  for G in groups
+      code = encode(G)
+      H = pc_group(order(G), code)
+      @test hom(G, H, gens(H)) isa Map
+      @test order(G) == order(H)
+      @test encode(H) == code
+  end
+end