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alg-02-10

# alg-02-10 - 2.3 COSETS AND THE THEOREM OF LAGRANGE 29 2.3...

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2.3. COSETS AND THE THEOREM OF LAGRANGE 29 2.3 Cosets and the Theorem of Lagrange We always assume that H is a subgroup of the group G . 2.3.1 Cosets Def 2.33. Let H be a subgroup of G . Given a G , the subset aH = { ah | h H } of G is the left coset of H containing a , while the subset Ha = { ha | h H } is the right coset of H containing a . Ex 2.34. H = eH = He is both a left coset and a right coset (In general aH 6 = Ha ). Properties of left cosets: (Similar for right cosets) 1. The number of elements in aH is equal to the number of elements in H . 2. If aH bH 6 = , then aH = bH . (Proof: If aH bH 6 = , then ah 1 = bh 2 for some h 1 ,h 2 H . So b = ah 1 h - 1 2 and bH = a ( h 1 h - 1 2 H ) = aH .) 3. From 2., G can be partitioned into left cosets of H . Prop 2.35 (Abelian Group). Let h G, + i be an abelian group and H G . Then 1. a + H = { a + h | h H } = { h + a | h H } = H + a. So “the left coset of H containing a ” = “the right coset of

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alg-02-10 - 2.3 COSETS AND THE THEOREM OF LAGRANGE 29 2.3...

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