Theorem-G satis es the right and left cancellation laws

Question

Let G be a group.

(i) G satises the right and left cancellation laws; that is, if a; b; x ≡ G, then ax = bx and xa = xb each imply that a = b.

(ii) If g ≡ G, then (g^-1)^-1 = g.

Answer 1

Proof:

(i) From ax = bx, we have axx^-1 = bxx^-1, then ae = be, then a = b. Similarly for the other case.

(ii) Temporarily denote the inverse of g^-1 by h (instead of (g^-1)^-1). Then the defining property of h, from the axiom for inverses applied to g^-1, is that

g^-1h = hg^-1 = e:

But g itself satises these equations in place of h, because the axiom for inverses applied to g says that

gg^-1 = g^-1g = e:

Hence, since inverses are unique, h = (g^-1)^-1 = g, as required.