Mathematics - Abstract Algebra - Groups and group theory - Group Axioms and Examples

10 steps: (1) A GROUP (G,*) is a set G with operation * satisfying: (2) CLOSURE: a,b∈G → a*b∈G. (3) ASSOCIATIVITY: (a*b)*c = a*(b*c). (4) IDENTITY: exists e∈G with a*e=e*a=a. (5) INVERSES: for each a∈G, exists a⁻¹ with a*a⁻¹=e. (6) Example: (ℤ,+). Identity=0, inverse of n is -n. (7) Example: (ℝ\\{0},