diff --git a/src/GrpAb/GrpAbFinGen.jl b/src/GrpAb/GrpAbFinGen.jl
index 3d5b350770..eb66476f9c 100644
--- a/src/GrpAb/GrpAbFinGen.jl
+++ b/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
diff --git a/test/GrpAb/GrpAbFinGen.jl b/test/GrpAb/GrpAbFinGen.jl
index 3e93fefd96..61dd2baf85 100644
--- a/test/GrpAb/GrpAbFinGen.jl
+++ b/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