Explanation Given information: H is the normal subgroup of group G . The alternating group G = A 4 H = { ( 1 ) , ( 1 , 2 ) ( 3 , 4 ) , ( 1 , 3 ) ( 2 , 4 ) , ( 1 , 4 ) ( 2 , 3 ) } Formula used: 1) If H is a normal subgroup of G then the group G / H that consists of the cosets of H in G is called the quotient group of G by H . G / H = { a H / a ∈ G } 2) If G is a finite group and H is the normal subgroup, then | G / H | = | G | | H | . Explanation: Elements in the alternating group, G = { ( 1 ) , ( 1 , 2 , 3 ) , ( 1 , 3 , 2 ) , ( 1 , 2 , 4 ) , ( 1 , 4 , 2 ) , ( 1 , 3 , 4 ) , ( 1 , 4 , 3 ) , ( 2 , 3 , 4 ) , ( 2 , 4 , 3 ) , ( 1 , 2 ) ( 3 , 4 ) , ( 1 , 3 ) ( 2 , 4 ) , ( 1 , 4 ) ( 2 , 3 ) } H = { ( 1 ) , ( 1 , 2 ) ( 3 , 4 ) , ( 1 , 3 ) ( 2 , 4 ) , ( 1 , 4 ) ( 2 , 3 ) } The quotient group is defined as G / H = { a H / a ∈ G } . G / H = { ( 1 ) H , ( 1 , 2 , 3 ) H , ( 1 , 3 , 2 ) H , ( 1 , 2 , 4 ) H , ( 1 , 4 , 2 ) H , ( 1 , 3 , 4 ) H , ( 1 , 4 , 3 ) H , ( 2 , 3 , 4 ) H , ( 2 , 4 , 3 ) H , ( 1 , 2 ) ( 3 , 4 ) H , ( 1 , 3 ) ( 2 , 4 ) H , ( 1 , 4 ) ( 2 , 3 ) H } The order of group G is 12 and H is 4 . Thus, the order of G / H = 12 / 4 = 3 Order of ( G / H ) = 3 . 1) First coset: ( 1 ) H = H 2) Second coset: ( 1 , 2 , 3 ) H = { ( 1 , 2 , 3 ) ( 1 ) , ( 1 , 2 , 3 ) ( 1 , 2 ) ( 3 , 4 ) , ( 1 , 2 , 3 ) ( 1 , 3 ) ( 2 , 4 ) , ( 1 , 2 , 3 ) ( 1 , 4 ) ( 2 , 3 ) } ( 1 , 2 , 3 ) ( 1 ) ⇒ [ 1 2 3 4 2 3 1 4 ] [ 1 2 3 4 1 2 3 4 ] ⇒ [ 1 2 3 4 1 2 3 4 ] ⇓ [ 1 2 3 4 2 3 1 4 ] ⇒ [ 1 2 3 4 2 3 1 4 ] ⇒ ( 1 , 2 , 3 ) ( 1 ) ( 1 , 2 , 3 ) ( 1 , 2 ) ( 3 , 4 ) ⇒ [ 1 2 3 4 2 3 1 4 ] [ 1 2 3 4 2 1 3 4 ] [ 1 2 3 4 1 2 4 3 ] ⇒ [ 1 2 3 4 1 2 4 3 ] ⇓ [ 1 2 3 4 2 1 3 4 ] ⇓ [ 1 2 3 4 2 3 1 4 ] ⇒ [ 1 2 3 4 3 2 4 1 ] ⇒ ( 1 , 3 , 4 ) Next ( 1 , 2 , 3 ) ( 1 , 3 ) ( 2 , 4 ) ⇒ [ 1 2 3 4 2 3 1 4 ] [ 1 2 3 4 3 2 1 4 ] [ 1 2 3 4 1 4 3 2 ] ⇒ [ 1 2 3 4 1 4 3 2 ] ⇓ [ 1 2 3 4 3 2 1 4 ] ⇓ [ 1 2 3 4 2 3 1 4 ] ⇒ [ 1 2 3 4 1 4 2 3 ] ⇒ ( 2 , 4 , 3 ) ( 1 , 2 , 3 ) ( 1 , 4 ) ( 2 , 3 ) ⇒ [ 1 2 3 4 2 3 1 4 ] [ 1 2 3 4 4 2 3 1 ] [ 1 2 3 4 1 3 2 4 ] ⇒ [ 1 2 3 4 1 3 2 4 ] ⇓ [ 1 2 3 4 4 2 3 1 ] ⇓ [ 1 2 3 4 2 3 1 4 ] ⇒ [ 1 2 3 4 4 1 3 2 ] ⇒ ( 1 , 4 , 2 ) ( 1 , 2 , 3 ) H = { ( 1 , 2 , 3 ) , ( 1 , 3 , 4 ) , ( 2 , 4 , 3 ) , ( 1 , 4 , 2 ) } (3) Third coset ( 1 , 3 , 2 ) H = { ( 1 , 3 , 2 ) ( 1 ) , ( 1 , 3 , 2 ) ( 1 , 2 ) ( 3 , 4 ) , ( 1 , 3 , 2 ) ( 1 , 3 ) ( 2 , 4 ) , ( 1 , 3 , 2 ) ( 1 , 4 ) ( 2 , 3 ) } ( 1 , 3 , 2 ) ( 1 ) ⇒ [ 1 2 3 4 3 1 2 4 ] [ 1 2 3 4 1 2 3 4 ] ⇒ [ 1 2 3 4 1 2 3 4 ] ⇓ [ 1 2 3 4 3 1 2 4 ] ⇒ , [ 1 2 3 4 3 1 2 4 ] ⇒ ( 1 , 3 , 2 ) ( 1 , 3 , 2 ) ( 1 , 2 ) ( 3 , 4 ) ⇒ [ 1 2 3 4 3 1 2 4 ] [ 1 2 3 4 2 1 3 4 ] [ 1 2 3 4 1 2 4 3 ] ⇒ [ 1 2 3 4 1 2 4 3 ] ⇓ [ 1 2 3 4 2 1 3 4 ] ⇓ [ 1 2 3 4 3 1 2 4 ] ⇒ [ 1 2 3 4 1 3 4 2 ] ⇒ ( 2 , 3 , 4 ) Now ( 1 , 3 , 2 ) ( 1 , 3 ) ( 2 , 4 ) ⇒ [ 1 2 3 4 3 1 2 4 ] [ 1 2 3 4 3 2 1 4 ] [ 1 2 3 4 1 4 3 2 ] ⇒ [ 1 2 3 4 1 4 3 2 ] ⇓ [ 1 2 3 4 3 2 1 4 ] ⇓ [ 1 2 3 4 3 1 2 4 ] ⇒ [ 1 2 3 4 2 4 3 1 ] ⇒ ( 1 , 2 , 4 ) ( 1 , 3 , 2 ) ( 1 , 4 ) ( 2 , 3 ) ⇒ [ 1 2 3 4 3 1 2 4 ] [ 1 2 3 4 4 2 3 1 ] [ 1 2 3 4 1 3 2 4 ] ⇒ [ 1 2 3 4 1 3 2 4 ] ⇓ [ 1 2 3 4 4 2 3 1 ] ⇓ [ 1 2 3 4 3 1 2 4 ] ⇒ [ 1 2 3 4 4 2 1 3 ] ⇒ ( 1 , 4 , 3 ) Thus, ( 1 , 3 , 2 ) H = { ( 1 , 3 , 2 ) , ( 2 , 3 , 4 ) , ( 1 , 2 , 4 ) , ( 1 , 4 , 3 ) } 4) Fourth coset: ( 1 , 2 , 4 ) H = { ( 1 , 2 , 4 ) ( 1 ) , ( 1 , 2 , 4 ) ( 1 , 2 ) ( 3 , 4 ) , ( 1 , 2 , 4 ) ( 1 , 3 ) ( 2 , 4 ) , ( 1 , 2 , 4 ) ( 1 , 4 ) ( 2 , 3 ) } ( 1 , 2 , 4 ) ( 1 ) ⇒ [ 1 2 3 4 2 4 3 1 ] [ 1 2 3 4 1 2 3 4 ] ⇒ [ 1 2 3 4 1 2 3 4 ] ⇓ [ 1 2 3 4 2 4 3 1 ] ⇒ , [ 1 2 3 4 2 4 3 1 ] ⇒ ( 2 , 4 , 1 ) ( 1 , 2 , 4 ) ( 1 , 2 ) ( 3 , 4 ) ⇒ [ 1 2 3 4 2 4 3 1 ] [ 1 2 3 4 2 1 3 4 ] [ 1 2 3 4 1 2 4 3 ] ⇒ [ 1 2 3 4 1 2 4 3 ] ⇓ [ 1 2 3 4 2 1 3 4 ] ⇓ [ 1 2 3 4 2 4 3 1 ] ⇒ [ 1 2 3 4 4 2 1 3 ] ⇒ ( 1 , 4 , 3 ) Next ( 1 , 2 , 4 ) ( 1 , 3 ) ( 2 , 4 ) ⇒ [ 1 2 3 4 2 4 3 1 ] [ 1 2 3 4 3 2 1 4 ] [ 1 2 3 4 1 4 3 2 ] ⇒ [ 1 2 3 4 1 4 3 2 ] ⇓ [ 1 2 3 4 3 2 1 4 ] ⇓ [ 1 2 3 4 2 4 3 1 ] ⇒ [ 1 2 3 4 3 1 2 4 ] ⇒ ( 1 , 3 , 2 ) ( 1 , 2 , 4 ) ( 1 , 4 ) ( 2 , 3 ) ⇒ [ 1 2 3 4 2 4 3 1 ] [ 1 2 3 4 4 2 3 1 ] [ 1 2 3 4 1 3 2 4 ] ⇒ [ 1 2 3 4 1 3 2 4 ] ⇓ [ 1 2 3 4 4 2 3 1 ] ⇓ [ 1 2 3 4 2 4 3 1 ] ⇒ [ 1 2 3 4 1 3 4 2 ] ⇒ ( 2 , 3 , 4 ) Thus, ( 1 , 2 , 4 ) H = { ( 2 , 4 , 1 ) , ( 1 , 4 , 3 ) , ( 1 , 3 , 2 ) , ( 2 , 3 , 4 ) } 5) Fifth coset: ( 1 , 4 , 2 ) H = { ( 1 , 4 , 2 ) ( 1 ) , ( 1 , 4 , 2 ) ( 1 , 2 ) ( 3 , 4 ) , ( 1 , 4 , 2 ) ( 1 , 3 ) ( 2 , 4 ) , ( 1 , 4 , 2 ) ( 1 , 4 ) ( 2 , 3 ) } ( 1 , 4 , 2 ) ( 1 ) ⇒ [ 1 2 3 4 4 1 3 2 ] [ 1 2 3 4 1 2 3 4 ] ⇒ [ 1 2 3 4 1 2 3 4 ] ⇓ [ 1 2 3 4 4 1 3 2 ] ⇒ , [ 1 2 3 4 4 1 3 2 ] ⇒ ( 1 , 4 , 2 ) ( 1 , 4 , 2 ) ( 1 , 2 ) ( 3 , 4 ) ⇒ [ 1 2 3 4 4 1 3 2 ] [ 1 2 3 4 2 1 3 4 ] [ 1 2 3 4 1 2 4 3 ] ⇒ [ 1 2 3 4 1 2 4 3 ] ⇓ [ 1 2 3 4 2 1 3 4 ] ⇓ [ 1 2 3 4 4 1 3 2 ] ⇒ [ 1 2 3 4 1 4 2 3 ] ⇒ ( 2 , 4 , 3 ) Next ( 1 , 4 , 2 ) ( 1 , 3 ) ( 2 , 4 ) ⇒ [ 1 2 3 4 4 1 3 2 ] [ 1 2 3 4 3 2 1 4 ] [ 1 2 3 4 1 4 3 2 ] ⇒ [ 1 2 3 4 1 4 3 2 ] ⇓ [ 1 2 3 4 3 2 1 4 ] ⇓ [ 1 2 3 4 4 1 3 2 ] ⇒ [ 1 2 3 4 3 2 4 1 ] ⇒ ( 1 , 3 , 4 ) ( 1 , 4 , 2 ) ( 1 , 4 ) ( 2 , 3 ) ⇒ [ 1 2 3 4 4 1 3 2 ] [ 1 2 3 4 4 2 3 1 ] [ 1 2 3 4 1 3 2 4 ] ⇒ [ 1 2 3 4 1 3 2 4 ] ⇓ [ 1 2 3 4 4 2 3 1 ] ⇓ [ 1 2 3 4 4 1 3 2 ] ⇒ [ 1 2 3 4 2 3 1 4 ] ⇒ ( 1 , 2 , 3 ) Thus, ( 1 , 4 , 2 ) H = { ( 1 , 4 , 2 ) , ( 2 , 4 , 3 ) , ( 1 , 3 , 4 ) , ( 1 , 2 , 3 ) } 6) Sixth coset: ( 1 , 3 , 4 ) H = { ( 1 , 3 , 4 ) ( 1 ) , ( 1 , 3 , 4 ) ( 1 , 2 ) ( 3 , 4 ) , ( 1 , 3 , 4 ) ( 1 , 3 ) ( 2 , 4 ) , ( 1 , 3 , 4 ) ( 1 , 4 ) ( 2 , 3 ) } ( 1 , 3 , 4 ) ( 1 ) ⇒ [ 1 2 3 4 3 2 4 1 ] [ 1 2 3 4 1 2 3 4 ] ⇒ [ 1 2 3 4 1 2 3 4 ] ⇓ [ 1 2 3 4 3 2 4 1 ] ⇒ [ 1 2 3 4 3 2 4 1 ] ⇒ ( 1 , 3 , 4 ) ( 1 , 3 , 4 ) ( 1 , 2 ) ( 3 , 4 ) ⇒ [ 1 2 3 4 3 2 4 1 ] [ 1 2 3 4 2 1 3 4 ] [ 1 2 3 4 1 2 4 3 ] ⇒ [ 1 2 3 4 1 2 4 3 ] ⇓ [ 1 2 3 4 2 1 3 4 ] ⇓ [ 1 2 3 4 3 2 4 1 ] ⇒ [ 1 2 3 4 2 3 1 4 ] ⇒ ( 1 , 2 , 3 ) Now ( 1 , 3 , 4 ) ( 1 , 3 ) ( 2 , 4 ) ⇒ [ 1 2 3 4 3 2 4 1 ] [ 1 2 3 4 3 2 1 4 ] [ 1 2 3 4 1 4 3 2 ] ⇒ [ 1 2 3 4 1 4 3 2 ] ⇓ [ 1 2 3 4 3 2 1 4 ] ⇓ [ 1 2 3 4 3 2 4 1 ] ⇒ [ 1 2 3 4 4 1 3 2 ] ⇒ ( 1 , 4 , 2 ) ( 1 , 3 , 4 ) ( 1 , 4 ) ( 2 , 3 ) ⇒ [ 1 2 3 4 3 2 4 1 ] [ 1 2 3 4 4 2 3 1 ] [ 1 2 3 4 1 3 2 4 ] ⇒ [ 1 2 3 4 1 3 2 4 ] ⇓ [ 1 2 3 4 4 2 3 1 ] ⇓ [ 1 2 3 4 3 2 4 1 ] ⇒ [ 1 2 3 4 1 4 2 3 ] ⇒ ( 2 , 4 , 3 ) Thus ( 1 , 3 , 4 ) H = { ( 1 , 3 , 4 ) , ( 1 , 2 , 3 ) , ( 1 , 4 , 2 ) , ( 2 , 4 , 3 ) } 7) Seventh coset : ( 1 , 4 , 3 ) H = { ( 1 , 4 , 3 ) ( 1 ) , ( 1 , 4 , 3 ) ( 1 , 2 ) ( 3 , 4 ) , ( 1 , 4 , 3 ) ( 1 , 3 ) ( 2 , 4 ) , ( 1 , 4 , 3 ) ( 1 , 4 ) ( 2 , 3 ) } ( 1 , 4 , 3 ) ( 1 ) ⇒ [ 1 2 3 4 4 2 1 3 ] [ 1 2 3 4 1 2 3 4 ] ⇒ [ 1 2 3 4 1 2 3 4 ] ⇓ [ 1 2 3 4 4 2 1 3 ] ⇒ [ 1 2 3 4 4 2 1 3 ] ⇒ ( 1 , 4 , 3 ) ( 1 , 3 , 4 ) ( 1 , 2 ) ( 3 , 4 ) ⇒ [ 1 2 3 4 4 2 1 3 ] [ 1 2 3 4 2 1 3 4 ] [ 1 2 3 4 1 2 4 3 ] ⇒ [ 1 2 3 4 1 2 4 3 ] ⇓ [ 1 2 3 4 2 1 3 4 ] ⇓ [ 1 2 3 4 4 2 1 3 ] ⇒ [ 1 2 3 4 2 4 3 1 ] ⇒ ( 1 , 2 , 4 ) Now ( 1 , 4 , 3 ) ( 1 , 3 ) ( 2 , 4 ) ⇒ [ 1 2 3 4 4 2 1 3 ] [ 1 2 3 4 3 2 1 4 ] [ 1 2 3 4 1 4 3 2 ] ⇒ [ 1 2 3 4 1 4 3 2 ] ⇓ [ 1 2 3 4 3 2 1 4 ] ⇓ [ 1 2 3 4 4 2 1 3 ] ⇒ [ 1 2 3 4 1 3 4 2 ] ⇒ ( 2 , 3 , 4 ) ( 1 , 4 , 3 ) ( 1 , 4 ) ( 2 , 3 ) ⇒ [ 1 2 3 4 4 2 1 3 ] [ 1 2 3 4 4 2 3 1 ] [ 1 2 3 4 1 3 2 4 ] ⇒ [ 1 2 3 4 1 3 2 4 ] ⇓ [ 1 2 3 4 4 2 3 1 ] ⇓ [ 1 2 3 4 4 2 1 3 ] ⇒ [ 1 2 3 4 3 1 2 4 ] ⇒ ( 1 , 3 , 2 ) Thus ( 1 , 4 , 3 ) H = { ( 1 , 4 , 3 ) , ( 1 , 2 , 4 ) , ( 2 , 3 , 4 ) , ( 1 , 3 , 2 ) } 8) Eighth coset ( 2 , 3 , 4 ) H = { ( 2 , 3 , 4 ) ( 1 ) , ( 2 , 3 , 4 ) ( 1 , 2 ) ( 3 , 4 ) , ( 2 , 3 , 4 ) ( 1 , 3 ) ( 2 , 4 ) , ( 2 , 3 , 4 ) ( 1 , 4 ) ( 2 , 3 ) } ( 2 , 3 , 4 ) ( 1 ) ⇒ [ 1 2 3 4 1 3 4 2 ] [ 1 2 3 4 1 2 3 4 ] ⇒ [ 1 2 3 4 1 2 3 4 ] ⇓ [ 1 2 3 4 1 3 4 2 ] ⇒ [ 1 2 3 4 1 3 4 2 ] ⇒ ( 2 , 3 , 4 ) ( 2 , 3 , 4 ) ( 1 , 2 ) ( 3 , 4 ) ⇒ [ 1 2 3 4 1 3 4 2 ] [ 1 2 3 4 2 1 3 4 ] [ 1 2 3 4 1 2 4 3 ] ⇒ [ 1 2 3 4 1 2 4 3 ] ⇓ [ 1 2 3 4 2 1 3 4 ] ⇓ [ 1 2 3 4 1 3 4 2 ] ⇒ [ 1 2 3 4 3 1 2 4 ] ⇒ ( 1 , 3 , 2 ) Now ( 2 , 3 , 4 ) ( 1 , 3 ) ( 2 , 4 ) ⇒ [ 1 2 3 4 1 3 4 2 ] [ 1 2 3 4 3 2 1 4 ] [ 1 2 3 4 1 4 3 2 ] ⇒ [ 1 2 3 4 1 4 3 2 ] ⇓ [ 1 2 3 4 3 2 1 4 ] ⇓ [ 1 2 3 4 1 3 4 2 ] ⇒ [ 1 2 3 4 4 2 1 3 ] ⇒ ( 1 , 4 , 3 ) ( 2 , 4 , 3 ) ( 1 , 4 ) ( 2 , 3 ) ⇒ [ 1 2 3 4 1 3 4 2 ] [ 1 2 3 4 4 2 3 1 ] [ 1 2 3 4 1 3 2 4 ] ⇒ [ 1 2 3 4 1 3 2 4 ] ⇓

8th Edition

ISBN: 9781285965918

Author: Gilbert

Publisher: Cengage

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Chapter 4.6, Problem 5E

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To find: The order of the quotient group G/H, the distinct element of G/H, and construct a multiplication table for G/H.

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Chapter 4.6, Problem 4E

Chapter 4.6, Problem 6E

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The order of the quotient group G / H , the distinct element of G / H , and construct a multiplication table for G / H .

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To find: The order of the quotient group G/H, the distinct element of G/H, and construct a multiplication table for G/H.

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