Let 7 and σ be elements of S5 such that: o(1) 2,0 (2) = 1, 0(3) = 4,0 (4) = 5,0 (5) = 3; o(1) = 5,0 (2) = 2, o(3) = 4, 0(4) = 3,0 (5) = 1. Calculate: (a) 100-100 (b) (0-172)33 (c) The order of the cyclic subgroup of S5 generated by 27³. (The order of a group is the number of elements in the group.)

Q: The subgroups of Z under addition are the groups nZ under addition for n. True or False then why

A: True or False   The subgroups of Z under addition are the groups nZ under addition for n.

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Q: How many cyclic subgroups does U(15) have?

A: To find Number of  cyclic subgroups does U(15) have

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A: We use the Cayley table.

Q: 5. Find the right cosets of the subgroup H in G for H = {(0,0), (1,0), (2,0)} in Z3 × Z2.

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Q: In Z, find all generators of the subgroup <10>  <12>.

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A: To show that a group of order 700 has a normal subgroup of order 175, we will make use of the Sylow…

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A: Let’s assume that the H is a subgroup of S5. So,

Q: Prove that a group of class equation 24 = 1+1+4+4+4+4+6 does not have a normal subgroup of order 4.

A: Prove that a group of class equation 24 = 1+1+4+4+4+4+6 does not have a normal subgroup of order 4.

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Q: To which group is Z12/ ker f isomorphic to? (a) Z₁ (b) V4 (c) D4 (d) A₁

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Q: Find the order of the factor group U(16)/(9)where U(16)={1,3,5,7,9,11,13,15} with operation…

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Q: 17*. Find an explicit epimorphism from A5 onto a group of order 3

A: Epimorphism: A homomorphism which is surjective is called Epimorphism.

Q: 5. How many automorphisms does Klein's 4-group have?

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A: Result: Let G, * be a group and H is a subset. The subset H is said to be subgroup, if for every a,…

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A: The objective of this question is to prove that Uk(n) is a subgroup of U(n). Here, U(n) represents…