I need an explanation, why
$ (\mathbb{Z}_7,\oplus _6 )$ is not a Group?
As I have discovered so far. The following conditions are satisfied
I) Closed!
II) Associative!
III) There exists identity element!!
IV) There exists inverse!!
Where is my mistake?
On the other hand I realized that there are some awkward results like the identity element in that group isn't unique.
There is no identity element. If you think that $0$ is it, compute $0\oplus_6 6$.