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(Recall the following notations: (a) is the cyclic group generated by the element a in a given group G; An (the alternating group) is the set of even permutations, on n objects; D₁ is the group of symmetries of a square; and T is the group of complex numbers with modulus 1, under the operation of multiplication.) (a) (8) in Z24 (b) (3) in U (8) (f) A₁ in S₁ (*Hint*) /*TT:.

(Recall the following notations: (a) is the cyclic group generated by the element a in a given group G; An (the alternating group) is the set of even permutations, on n objects; D₁ is the group of symmetries of a square; and T is the group of complex numbers with modulus 1, under the operation of multiplication.) (a) (8) in Z24 (b) (3) in U (8) (f) A₁ in S₁ (*Hint*) /*TT:.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter3: Groups
Section3.4: Cyclic Groups
Problem 8E: Exercises 8. Let be an element of order in a group. Find the order of each of the following. a. ...
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Please do part A and B and please show step by step and explain

Exercise 18.1.7. List the left and right cosets of the subgroups in each of the following. Tell whether the left and right cosets are equal.
Transcribed Image Text:Exercise 18.1.7. List the left and right cosets of the subgroups in each of the following. Tell whether the left and right cosets are equal.
(Recall the following notations: (a) is the cyclic group generated by the element a in a given group G; An (the alternating group) is the set of even permutations, on n objects; D₁ is the group of symmetries of a square; and T is the group of complex numbers with modulus 1, under the operation of multiplication.) (a) (8) in Z24 (b) (3) in U (8) (c) 4Z in Z (d) H = {(1), (123), (132)} in S₁ (e) H = {1, i,-1,-i} in Q8 (See Ex- ample 15.2.8) (f) A4 in S4 (*Hint*) (g) An in Sn (*Hint*) (h) D₁ in S₁ (*Hint*) (i) T in C*
Transcribed Image Text:(Recall the following notations: (a) is the cyclic group generated by the element a in a given group G; An (the alternating group) is the set of even permutations, on n objects; D₁ is the group of symmetries of a square; and T is the group of complex numbers with modulus 1, under the operation of multiplication.) (a) (8) in Z24 (b) (3) in U (8) (c) 4Z in Z (d) H = {(1), (123), (132)} in S₁ (e) H = {1, i,-1,-i} in Q8 (See Ex- ample 15.2.8) (f) A4 in S4 (*Hint*) (g) An in Sn (*Hint*) (h) D₁ in S₁ (*Hint*) (i) T in C*
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