Proofs from Group Theory

Cancellation Law: Let G be a group such that a, b, c ∈ G. If a∗c = b∗c, then a = b. Proof Suppose a ∗ c = b ∗ c. Since c ∈ G, it follows that an element d exists such that c ∗ d = e. Now, if we multiply both sides by d on the right, we obtain (a ∗ c) ∗ d = (b ∗ c) ∗ d. By associativity, we obtain a ∗ (c ∗d) = b ∗ (c ∗d). Since c ∗d = e, we obtain a ∗ e = b ∗ e, and by de nition of identity element, we obtain a = b as desired.