MATH 113 Lecture Notes - Lecture 5: If And Only If, Cyclic Group
Document Summary
Lemma: for every x, x * e = x. X" * x * e = x" * z. E = x" * z x" * z = e. By lemma 1 ( a*b = a*c -> b = c), z = x. Suppose e and e" s. t. both satisfy (ii) (e*a = a) e * e" = e" by (ii), and e*e" = e by lemma. Lemma: for every x, x * x" = e. Suppose z s. t. x * x" = z. We show that z = e. (x" x) * x" = x" (x * x") = x" * z. Also by lemma, x" * e = x". X" * e = x" * z, so e = z. Corollary: the inverse of a is unique there is only one inverse to a given element. Theorem: h is a subgroup of g iff.
