Let (S, *) be a semigroup with identity e. Suppose x S and suppose that x^-1 exists in (S, *).

By definition (2) of Powers :

x^(n+1) = x^n * x n 1 (A)

Need to show by definitions 1 - 4 that (A) is true for all "n" that is an element of an integer in the case that x^-1 exists.

Definition 1 - 4: Let (S, *) be a semigroup. Let x S

(1) x^1 = x; 1 positive integer

(2) x^(n+1) = x^n * x n 1

(3) If (S, *) has an identity e, then, x^o = e; o positive integer

(4) If x has a inverse x S, then, x^-n = (x)^n , n positive integer

Hint: 1st: show (A) is true for n = 0 and n = -1

2nd: Let n = - k, when k 2

Show x ^(n+1) and show x^n *x both equal something, then conclude (A) is true n integer.