1, off (obvious)
2, the legal connection
(a * b)* c =(a+b-2)* c = a+b-2+c-2 = a+b+c-4
a *(b * c)= a *(b+ c-2)= a+b+ c-2-2 = a+b+ c-4
Then (a*b)*c=a*(b*c)
3. The unit element exists, which is 2, because a * 2 = 2 * a = a
There is an inverse element, a=4-a, because a*(4-a)=2.
Question 6
Obviously, the identity element is an idempotent element of a group.
By reducing to absurdity, suppose there is a non-unit element a (a≠e, e is a unit element), which is also an idempotent element in a group.
And then a? =a
Multiply both sides of the equation by a at the same time and you get
Answer? *a=a*a
Answer? *a=e
namely
a*(a*a)=e
therefore
a*e=e
that is
a=e
This contradicts the assumption of a≠e, so the idempotents in a group are unique.