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: The elements of the quotient group (/18) are 1- 2- {.-208,-108,008,108,208,.} 3- (0,1,2,3, .8) 4-…

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Q: A group G is cyclic if and only if there exists at G such that G={a^|n  ∈ Z}. True or False then why

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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: 3. A ring are defined busing THREE (3) axioms that need to satisfy such as Abelian Group,…

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Q: Prove that if x is a group element with infinite order, then x^m is not equal to x^n when m is not…

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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: Write under isomorphism all the abelian groups of order 8 and their respective subgroups.

A: To find all the abelian groups of order 8, we can use the fundamental theorem of finitely generated…

Q: Without using the 2 . Odd Test, prove that there is no simple groupof order 210.

A: In this question, we have to show that there is no simple group of order 210.

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: 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: 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: Give the order of the element in the factor group (2,3)+⟨(0,3)⟩ in (Z10×Z4)/⟨(0,3)⟩

A: The order of the element (2 3 +  <(0, 3)> in (Z⁠⁠⁠⁠⁠⁠⁠10×Z4) / <(0, 3)> is…

Q: Let G be a group and suppose that a * b * c = e. Show that b * c *a = e.

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Q: Let G be a group and a ∈ G. The centralizer of  a in G is equal to the centralizer of a^-1 in G.…

A: First let us see the definition of centralizer or normalizer of a in G (definition from I.N.…

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

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Q: Find the index of in the group Z24

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Q: Q3\Prove that if (G,*) be a finite group of prime order then (G,*) is an abelian group.

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Q: 2. Use one of the Subgroups Tests from Chapter 3 to prove that when G is an Abelian group and when n…

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Q: Let f and g be permutations on the set {1, 2, 3, 4, 5, 6, 7}, defined as follows (b) f 1 (c) fgf =…

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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: 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: x and y are elements of group G, prove |x| = |g^-1xg|. G is not abelian

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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: The elements of the quotient group (2/8). 12 (0,1,2,3,...,8) 1- 2- -208,-108,008,108,208,...) 3- 4-…

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