Recall that the symmetric group S3 of degree 3 is the group of all permuations on the set {1, 2, 3} and its elements can be listed in the cycle form as follows: S3 = {(), (1 2), (1 3), (2 3), (1 2 3), (1 3 2)} Let 9: S3 + {-1, 1} be the group hornomorphism given by the rule: -1 if f is an odd permutation p (f) = 1 if f is an even permutation a. Compute p ((1 2 3)). ((1 2 3)) = b. Compute the range Ran () of the group homomorphism p. Ran (y) = c. Compute the size of the kernel Ker (y) of the group homomorphism p. |Ker (y)| = d. Is p an isomorphism?

Recall that the symmetric group S3 of degree 3 is the group of all permuations on the set {1, 2, 3} and its elements can be listed in the cycle form as follows: S3 = {(), (1 2), (1 3), (2 3), (1 2 3), (1 3 2)} Let 9: S3 + {-1, 1} be the group hornomorphism given by the rule: -1 if f is an odd permutation p (f) = 1 if f is an even permutation a. Compute p ((1 2 3)). ((1 2 3)) = b. Compute the range Ran () of the group homomorphism p. Ran (y) = c. Compute the size of the kernel Ker (y) of the group homomorphism p. |Ker (y)| = d. Is p an isomorphism?

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter3: Groups

Section3.5: Isomorphisms

Problem 5E

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