Proof synopsis We give an example of a proof synopsis. Here is a one-sentence synopsis of the proof that the inverse of an element a in a group (G, *) is unique. Assuming that a * a' = e and a * a" = e, apply the left cancellation law to the equation a * a' = a * a". Note that we said "the left cancellation law" and not "Theorem 4.15." We always suppose that our synopsis was given as an explanation given during a conversation at lunch, with no reference to text numbering and as little notation as is practical.

Give a one-sentence synopsis of the proof of the left cancellation law in Theorem 4.15.

Data from Theorem 4.15

If G is a group with binary operation *, then the left and right cancellation laws hold in G, that is, a * b = a * c implies b = c, and b * a = c * a implies b = c for all a, b, c ∈ G.

Proof

Suppose a * b =a * c. Then by G₃ there exists a', and a' * (a* b) =a' * (a * c). 

By the associative law, (a' * a) * b = (a' * a) * c. 

By the definition of a' in G₃ a' * a= e, so e * b = e * c. 

By the definition of e in G₂  b = C. 

Similarly, from b * a = c * a one can deduce that b = c upon multiplication on the right by a' and use of the axioms for a group.