feat: add isomorphism method for abelian groups (#1724)

Author: thofma

src/GrpAb/GrpAbFinGen.jl

@@ -606,6 +606,14 @@ function is_isomorphic(G::FinGenAbGroup, H::FinGenAbGroup)
   return elementary_divisors(G) == elementary_divisors(H)
 end
 
+function isomorphism(G::FinGenAbGroup, H::FinGenAbGroup)
+  SG, SGtoG = snf(G)
+  SH, SHtoH = snf(H)
+  @req SG.snf == SH.snf "Groups are not isomorphic"
+  h = hom(SG, SH, gens(SH))
+  return inv(SGtoG) * h * SHtoH
+end
+
 ################################################################################
 #
 #  Direct products/direct sums/biproducts

test/GrpAb/GrpAbFinGen.jl

@@ -176,10 +176,18 @@
   @testset "Isomorphism" begin
     b = @inferred is_isomorphic(abelian_group(Int[]), abelian_group(Int[]))
     @test b
+    h = isomorphism(abelian_group(Int[]), abelian_group(Int[]))
+    @test is_bijective(h)
 
     G = abelian_group([2, 3, 5])
     H = abelian_group([30])
     @test @inferred is_isomorphic(G, H)
+
+    h = isomorphism(G, H)
+    @test is_bijective(h)
+
+    K = abelian_group([2, 3])
+    @test_throws ArgumentError isomorphism(G, K)
   end
 
   @testset "Direct product" begin