Abstract Algebra Theory and Applications - Computer Science ...

5.1 COSETS 914. g 2 ∈ g 1 H;5. g −11 g 2 ∈ H.In all of our examples the cosets of a subgroup H partition the largergroup G. The following theorem proclaims that this will always be the case.Theorem 5.2 Let H be a subgroup of a group G. Then the left cosets ofH in G partition G. That is, the group G is the disjoint union of the leftcosets of H in G.Proof. Let g 1 H and g 2 H be two cosets of H in G. We must show thateither g 1 H ∩ g 2 H = ∅ or g 1 H = g 2 H. Suppose that g 1 H ∩ g 2 H ≠ ∅ anda ∈ g 1 H ∩ g 2 H. Then by the definition of a left coset, a = g 1 h 1 = g 2 h 2for some elements h 1 and h 2 in H. Hence, g 1 = g 2 h 2 h −11 or g 1 ∈ g 2 H. ByLemma 5.1, g 1 H = g 2 H.□Remark. There is nothing special in this theorem about left cosets. Rightcosets also partition G; the proof of this fact is exactly the same as the prooffor left cosets except that all group multiplications are done on the oppositeside of H.Let G be a group and H be a subgroup of G. Define the index of Hin G to be the number of left cosets of H in G. We will denote the indexby [G : H].Example 3. Let G = Z 6 and H = {0, 3}. Then [G : H] = 3.Example 4. Suppose that G = S 3 , H = {(1), (123), (132)}, and K ={(1), (12)}. Then [G : H] = 2 and [G : K] = 3. Theorem 5.3 Let H be a subgroup of a group G. The number of left cosetsof H in G is the same as the number of right cosets of H in G.Proof. Let L H and R H denote the set of left and right cosets of H inG, respectively. If we can define a bijective map φ : L H → R H , then thetheorem will be proved. If gH ∈ L H , let φ(gH) = Hg −1 . By Lemma 5.1,the map φ is well-defined; that is, if g 1 H = g 2 H, then Hg1 −1 = Hg2 −1 . Toshow that φ is one-to-one, suppose thatHg −11 = φ(g 1 H) = φ(g 2 H) = Hg −12 .Again by Lemma 5.1, g 1 H = g 2 H. The map φ is onto since φ(g −1 H) = Hg.□