add hom between GAPGroup and MultTableGroup`

src/Groups/homomorphisms.jl

@@ -133,6 +133,13 @@ function hom(G::GAPGroup, H::GAPGroup, imgs::Vector; check::Bool = true)
   return hom(G, H, gens(G), imgs; check)
 end
 
+
+################################################################################
+#
+#  `hom` between `GAPGroup` and `FinGenAbGroup`
+#
+################################################################################
+
 # Map `G::GAPGroup` to `A::FinGenAbGroup` by prescribing images.
 # Return a composition of homomorphisms `G -> G/G' -> B -> A`,
 # not a `GAPGroupHomomorphism`.
@@ -172,6 +179,45 @@ function hom(A::FinGenAbGroup, G::GAPGroup, imgs::Vector{<:GAPGroupElem}; check:
   return hom(A, G, gens(A), imgs; check)
 end
 
+################################################################################
+#
+#  `hom` between `GAPGroup` and `MultTableGroup`
+#
+################################################################################
+
+# Map `G::GAPGroup` to `M::MultTableGroup` by prescribing images.
+# Return a composition of homomorphisms `G -> B -> M`,
+# not a `GAPGroupHomomorphism`.
+function hom(G::GAPGroup, M::MultTableGroup, gensG::Vector, imgs::Vector{MultTableGroupElem}; check::Bool = true)
+  # map M to an isomorphic perm. group B
+  iso = isomorphism(PermGroup, M)
+  B = codomain(iso)
+
+  # map G to B as prescribed
+  map1 = hom(G, B, gensG, [iso(x) for x in imgs], check = check)
+
+  # create the composition
+  return compose(map1, inv(iso))
+end
+
+function hom(G::GAPGroup, M::MultTableGroup, imgs::Vector{MultTableGroupElem}; check::Bool = true)
+  return hom(G, M, gens(G), imgs; check)
+end
+
+# Map `M::MultTableGroup` to `G::GAPGroup` by prescribing images.
+# Return a composition `M -> B -> G` where `M -> B` is an isomorphism
+# to a `GAPGroup`.
+function hom(M::MultTableGroup, G::GAPGroup, gensM::Vector, imgs::Vector{<:GAPGroupElem}; check::Bool = true)
+  map1 = isomorphism(PermGroup, M)
+  map2 = hom(codomain(map1), G, [map1(x) for x in gensM], imgs, check = check)
+  return compose(map1, map2)
+end
+
+function hom(M::MultTableGroup, G::GAPGroup, imgs::Vector{<:GAPGroupElem}; check::Bool = true)
+  return hom(M, G, gens(M), imgs; check)
+end
+
+
 function domain(f::GAPGroupHomomorphism)
   return f.domain
 end

test/Groups/homomorphisms.jl

@@ -334,12 +334,19 @@ end
       G  = Hecke.small_group(64, 14, DB = Hecke.DefaultSmallGroupDB())
       H = small_group(64, 14)
       @test is_isomorphic(G, H)
+      @test is_isomorphic(H, G)
       f = isomorphism(G, H)
       for x in gens(G), y in gens(G)
          @test f(x) * f(y) == f(x * y)
          @test preimage(f, f(x)) == x
          @test preimage(f, f(y)) == y
       end
+      f = isomorphism(H, G)
+      for x in gens(H), y in gens(H)
+         @test f(x) * f(y) == f(x * y)
+         @test preimage(f, f(x)) == x
+         @test preimage(f, f(y)) == y
+      end
       fl, f = is_isomorphic_with_map(G, H)
       @test fl
       for x in gens(G), y in gens(G)
@@ -636,6 +643,69 @@ end
    @test order(image(mp)[1]) == 1
 end
 
+@testset "Homomorphism GAPGroup to MultTableGroup" begin
+   # M isomorphic to G
+   M = Hecke.small_group(20, 3, DB = Hecke.DefaultSmallGroupDB())
+   iso = isomorphism(PermGroup, M)
+   G = codomain(iso)
+   imgs = [iso(x) for x in gens(M)]
+   mp = hom(G, M, imgs, gens(M))
+   @test [mp(x) for x in gens(G)] == gens(M)
+   @test [preimage(mp, x) for x in gens(M)] == gens(G)
+
+   # G trivial
+   G = cyclic_group(PcGroup, 1)
+   M = Hecke.small_group(6, 1, DB = Hecke.DefaultSmallGroupDB())
+   imgs = elem_type(M)[]
+   mp = hom(G, M, imgs)
+   @test mp(one(G)) == one(M)
+
+   # M trivial
+   G = small_group(20, 3)
+   M = Hecke.small_group(1, 1, DB = Hecke.DefaultSmallGroupDB())
+   imgs = [one(M) for x in gens(G)]
+   mp = hom(G, M, imgs)
+   @test all(x -> order(mp(x)) == 1, gens(G))
+
+   # G and M trivial
+   G = cyclic_group(PcGroup, 1)
+   M = Hecke.small_group(1, 1, DB = Hecke.DefaultSmallGroupDB())
+   imgs = elem_type(M)[]
+   mp = hom(G, M, imgs)
+   @test mp(one(G)) == one(M)
+end
+
+@testset "Homomorphism MultTableGroup to GAPGroup" begin
+   # M isomorphic to G
+   M = Hecke.small_group(20, 3, DB = Hecke.DefaultSmallGroupDB())
+   iso = isomorphism(PermGroup, M)
+   G = codomain(iso)
+   imgs = [iso(x) for x in gens(M)]
+   mp = hom(M, G, gens(M), imgs)
+   @test [mp(x) for x in gens(M)] == gens(G)
+
+   # G trivial
+   G = cyclic_group(PcGroup, 1)
+   M = Hecke.small_group(6, 1, DB = Hecke.DefaultSmallGroupDB())
+   imgs = [one(G) for x in gens(M)]
+   mp = hom(M, G, imgs)
+   @test mp(one(M)) == one(G)
+
+   # M trivial
+   M = Hecke.small_group(1, 1, DB = Hecke.DefaultSmallGroupDB())
+   G = dihedral_group(8)
+   imgs = [one(G)]
+   mp = hom(M, G, imgs)
+   @test mp(one(M)) == one(G)
+
+   # G and M trivial
+   M = Hecke.small_group(1, 1, DB = Hecke.DefaultSmallGroupDB())
+   G = cyclic_group(PcGroup, 1)
+   imgs = [one(G)]
+   mp = hom(M, G, imgs)
+   @test mp(one(M)) == one(G)
+end
+
 function test_direct_prods(G1,G2)
    G = direct_product(G1,G2)
    f1 = canonical_injection(G,1)