Group Theory

Walkthrough

Verify that (ℤ, +) is a group.

1. 1.Recall the four group axioms: closure, associativity, identity, and inverses.
2. 2.Closure: for any a, b ∈ ℤ, a + b ∈ ℤ. ✓
3. 3.Associativity: (a + b) + c = a + (b + c) for all a, b, c ∈ ℤ. ✓
4. 4.Identity: 0 + a = a for all a ∈ ℤ, so 0 is the identity element. ✓
5. 5.Inverse: for any a ∈ ℤ, a + (−a) = 0, so −a is the inverse of a. ✓ All four axioms hold, so (ℤ, +) is a group.