If cyclic group G is generated by an element  a of order n, then a\power{m} is a generator of G iff  (m,m)=1 i.e., the greatest common divisor  of m and n is 1

Q: ication modulo 4 is not a

A: Show that {1, 2, 3} under multiplication modulo 4 is not a group but that {1, 2, 3, 4} under…

Q: let G=3^n, n is all integers. show that multiplication is an operation on G and show that is an…

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Q: Find all abelian groups (upto isomorphism) of order 3375.

A: We have to find the all abelian groups (upto isomorphism) of order 3375.

Q: 9) Let H be a subgroup of a group G and a, be G. Then a e bH if and only if O b-la e H O ba e H O…

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Q: For any subset H of a group G, if ab^-1 is in H for all a,b element of H then H is a subgroup of G.…

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Q: An element is called a square if it can be expressed in the form b2for some b. Suppose that G is an…

A: Given: An element is called a square if it can be expressed in the form b2 for some b.G is an…

Q: Consider from the b the Symmetric group. 1 2 3 5 6 7 7 5 4 1 2 36 3 5 6 = 3 7 15 a) Express each of…

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Q: Prove that a subgroup of a finite abelian group is abelian. Be careful when checking the required…

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Q: group ⟨a,b,c|b^4 = a,c^(−1) = b⟩ is abelian.

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Q: How many elements of a cyclic group with order 10 have order dividing 5?

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Q: Let G be a group and a ∈ G. Suppose that |a| = 18, find |a^21|. Suppose that |a| = 32, find a…

A: We are given that G is a group, and a ∈ G.A) The order of element a is given as 18. a=18 ⇒a18 = e,…

Q: 1- The index 2- infinite group 3- permute elements B prove that (cent G,*) is normal subgroup of the…

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Q: Computations In Exercises 1 through 6, determine whether the binary operation gives a group…

A: As per the policy we are solving only 3 subparts of the given question. Please repost the question…

Q: Show that a group in which all the mth powers commute with each other and all the nth powers commute…

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Q: Let G1 be a group defined on the set Z4 with binary relation, +, addition modulo 4, and let G2 be a…

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Q: *Prove: Every subgroup of a cyclic group is cyclic.

A: Let G be a cyclic group generated by an element a. That is, G = ⟨a⟩. We want to prove that every…

Q: Solution (incorrect!) (i) Group G: Modular multiplication is commutative, so G is abelian. Hence, by…

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Q: Compute the center of generalized linear group for n=4

A: To find - Compute the center of generalized linear group for n=4

Q: How many elements of a cyclic group with order 14 have order 7?

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Q: Let p be a prime number and (G, *) a finite group IGI= p?. How can you prove that the group (G, *)…

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Q: Suppose that a is an element of a group G. Prove that if there is some integer n, n notequalto 0,…

A: Suppose that a is an element of a group G. Prove that if there is some integer n, such that n≠0 and…

Q: 9 Find all isamorphism classes of abelian groups of arder 72.

A: We have find isomorphism classes of abelian group of order 72:

Q: Alte

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Q: Consider the group D4 = (a, b) = {e = (1), a, a², a³, b, ab, a²b, a³b} %3D where a = (1 23 4) and b…

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Q: 6) Use the given lattices to help find the centralizers of every element in the following groups.…

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Q: 2. Is the set Z3 = {0,1,2} form a group with respect to addition modulo 3 how about to…

A: We use caley's table to verify the properties of a group.

Q: prove or disprove The intersection of any two distinct left cosets in the group is empty set

A: "Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: your a) G = (Z12,O); H = {0,2,4, 6, 8, 10} %3D

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