We need to test for closure, associativity, identity and inverse.
CLOSURE
a＊b=a+b-1 by definition. In the set of integers (ℤ), a+b is also an integer, because a, b in ℤ..
Let a+b=c in ℤ.. But -1 is also an integer so c-1 is in ℤ., therefore a＊b is in ℤ.. Closure applies.
ASSOCIATIVITY
If a＊(b＊c)=(a＊b)＊c then associativity applies. b＊c=b+c-1. a＊(b＊c)=a+b+c-1-1=a+b+c-2.
a＊b=a+b-1, so (a＊b)＊c=a+b-1+c-1=a+b+c-2. Associativity applies.
IDENTITY
The identity element is denoted by e.
a＊e=e＊a=a, so a+e-1=e+a-1=a, and e=1 is the identity element.
INVERSE
Let the inverse of a be ɐ, then a＊ɐ=e=1.
a＊ɐ=a+ɐ-1=1, so ɐ=2-a. The inverse of a is 2-a. So the inverse of ɐ should be a.
Let the inverse of ɐ be ɓ, then we should find ɐ＊ɓ=a:
(2-a)＊ɓ=2-a+ɓ-1=1, 2-a+ɓ=2, and ɓ=a, so the inverse property holds.
All four properties of a group have been satisfied so the ＊ operation between integers can be applied to the set of all integers. a＊b form a group.