MATH 113 Lecture Notes - Lecture 13: Coset, Bijection, If And Only If
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Recall the left coset is of g g is gh = {g h| h h} and h is a subgroup of g. a ~h b b-1 * a h . Lemma: gh = ah iff g ~h a (h is a subgroup of g, and g, a g) This means a-1 g h, which means there exists an h h so that a. This means that there exists an h h so that g = a*h. and gh = ahh = {a*h*h. This is a subset of {a*h" | h" h} = ah. By a symmetric argument (switching a, g), you can conclude that ah is a subset of gh. Eg = eh h, so g*e the same to say that a"g = h. h h, so finally a"g h, so a ~ This means there exists h h so that g * e.
