THEOREM. Let ?=be a cyclic group of order ?. Then am is a generator of ? if and only if ? and ? are relatively prime. 1- Use the theorem discussed above to find all the generators of Z7 under addition. 2- Use the theorem discussed above to find all the generators of U7 under multiplication

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A: We have to state whether the given statement is true or false : The given statement is : The…

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A: Use property of group and solve it.

Q: Show that any group of order 3 is abelian.

A: The solution is given as

Q: 5. Find the number of generators of the cyclic group Z15

A: To find the number of generators of the cyclic group ℤ15.

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Q: 1. Prove or disprove (a). The order of the element (2, 3) € Z6X Z15 is 5. (b). The group Z7X Z17 x…

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Q: 1ABCD E 1 A D E

A: Commutative group of order 6 is z6 under multiplication.

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A: The complete solutions are given below

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Q: . Let ϕ : Z31 → Z31 be a homomorphism such that ϕ(16) = 19. Calculate ϕ(1) and ϕ(10).

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Q: (a) Let G = {e, a, . .. , aº | a10 = e} be the cyclic group of order 10. For which m, is G = (am)?…

A: As per our guideline we are supposed to answer only first asked  question. Kindly repost other…

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Q: 18. An external direct product G₁ G₂ G₂of a finite number of cyclic groups is cyclic if and only if…

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Q: Which of the following groups are cyclic? For each cyclic group, list all the generators of the…

A: To identify the given group is cyclic or not.

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Q: create the addition and multiplication table in Z5 a) show that (Z5, +) is an abelian group b) show…

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A: We have to check

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Q: : Show that if G is a cyclic group then G is abelian. : Let n>0 be a positive integer, and let a and…

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Q: For any elements, a and b in a group G, a^(5).b = ba and a^(4) = e, then G is an abelian group.

A: Given a and b in a group G. and  a5b = ba  and  a4 = e To prove:- G is abelian group

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Q: The following is a Cayley table for a group G * 1 2 3 4 6. 7 8. 1 1 3 4 6. 8. 1 8. 4 3 3 6. 7 8. 1 2…

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Q: (1) Let G be a group of order 94 = 2 x 47. Let Z(G), as usual, denote the center of G. (a) For each…

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Q: Q 12 Every subgroup of an abelian group is abelian.

A: As per our guidelines we are allowed to do 1 question at a time. Please post other questions next…

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Q: Using the definition 4.1 and 4.3 of a group G1,G2 and G3 axioms to show that: a. Z is a group under…

A: Using the definition 4.1 and 4.3 of a group G1, G2 and G3 axioms to show that: (a) Z is a group…

Q: The group U(15) is an internal direct product of the cyclic subgroups generated by 7 by 11, U(15) =…

A: We  have to check that U(15)= <7>×<11>  Or not. Concept: If  n =p1.p2 Where p1 and p2…

Q: If I is an ideal of R, then by definition, (I, +) is an abelian group. Consequently, it has an…

A: Since I is ideal , therefore I is definitely a subset of the ring R.

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Q: Prove or disprove the following statements: (a) If (ab)n+1 = an+¹fn+¹ a, b = G, n=1,2,..., then G is…

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Q: (A) Prove that, every group of prime order is cyclic.

A: Let, G be a group of prime order. That is: |G|=p where p is a prime number.

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Q: (d) Prove that 1+ N is a group with respect to multiplication. (e) Verify that 1+N(Z27) is a cyclic…

A: d) To prove that 1+N is a group. Let a=1+n, b=1+n'∈1+N where n,n'∈N=Nℤn. Then,…

Q: Every element of a cyclic group generates the group. True or False then why

A: False Every element of cyclic group do not generate the group.