allow == for two groups/group elements in fewer cases

If `_check_compatible` for the two groups in question returns `false`
then throw an exception.
Note that the result is not based only on the types of the arguments
but also on the groups.

src/Groups/GAPGroups.jl

@@ -372,9 +372,26 @@ end
 
 Base.:*(x::GAPGroupElem, y::GAPGroupElem) = _prod(x, y)
 
-==(x::GAPGroup, y::GAPGroup) = GapObj(x) == GapObj(y)
+function ==(x::GAPGroup, y::GAPGroup)
+ _check_compatible(x, y; error = false) || throw(ArgumentError("x and y are not compatible"))
+ return GapObj(x) == GapObj(y)
+end
+
+# For two `BasicGAPGroupElem`s,
+# we allow the question for equality if their parents fit together
+# in the sense of `_check_compatible`,
+# and compare the `GapObj`s if this is the case.
+function ==(x::BasicGAPGroupElem, y::BasicGAPGroupElem)
+ _check_compatible(parent(x), parent(y); error = false) || throw(ArgumentError("parents of x and y are not compatible"))
+ return GapObj(x) == GapObj(y)
+end
 
-==(x::BasicGAPGroupElem, y::BasicGAPGroupElem ) = GapObj(x) == GapObj(y)
+# For two `GAPGroupElem`s,
+# if no specialized method is applicable then no `==` comparison is allowed.
+function ==(x::GAPGroupElem, y::GAPGroupElem)
+ _check_compatible(parent(x), parent(y); error = false) || throw(ArgumentError("parents of x and y are not compatible"))
+ throw(ArgumentError("== is not implemented for the given types"))
+end
 
 """
  one(G::GAPGroup) -> elem_type(G)

test/Groups/conformance.jl

@@ -60,6 +60,10 @@ include(joinpath(dirname(pathof(AbstractAlgebra)), "..", "test", "Groups-conform
  @test g>h || g==h || g<h
  @test isequal(g,h) || isless(g,h) || isless(h,g)
  end
+
+ F = free_group(1)
+ @test_throws ArgumentError F == G
+ @test_throws ArgumentError gen(F, 1) == g
  end
 
  @testset "Group operations" begin