《Discrete Mathematics II教学-华南理工》Section 10 Cosets and the Theorem of Lagrange.pdfVIP

II.10 Cosets and the Theorem of Lagrange 1 Section II.10. Cosets and the Theorem of Lagrange Note. In this section, we prove that the order of a subgroup of a given finite group divides the order of the group. This is called Lagrange’s Theorem. The proof involves partitioning the group into sets called cosets. Later, we will form a group using the cosets, called a factor group (see Section 14). Theorem 10.1. Let H be a subgroup of group G. Let the relation ∼L be defined on G by a ∼L b iff a−1b ∈ H. Let the relation ∼R be defined by a ∼R b iff ab−1 ∈ H. Then ∼L and ∼R are both equivalence relations on G. Definition 10.2. Let H be a subgroup of a group G. The subset aH = {ah | h ∈ H } of G is the left coset of H containing a. The subset Ha = {ha | h ∈ H } is the right coset of H containing a. Note. Suppose x, y ∈ aH . Then x = ah and y = ah for some h , h ∈ H . So 1 2 1 2 h1 = a−1x and h2 = a−1y . So a ∼L x and a ∼L y . Therefore, x ∼L y . Now e ∈ H since H is a group, so a = ae ∈ aH . Equivalently, a−1a = e ∈ H , so a ∼L a. So the coset aH is actually the ∼L equivalence class of elements of G which contains a. Similarly, Ha is the ∼R equivalence class of elements of G which contains a. II.10 Cosets and the Theorem of Lagrange 2 Exercise 10.4. Find the cosets of the subgroup 4 of Z12 . Solution. First, 4 = {0, 4, 8} and Z12 is an additive group. So we get the cosets: 0 + 4 = {0, 4, 8} = 4 + 0 1 + 4 = {1, 5, 9} = 4 + 1 2 + 4 = {2, 6, 10} = 4 + 2