Prove C*, the group of nonzero complex numbers under multiplication, has a cyclic subgroup of order "n" for every positive integer "n"

Q: Prove that a group of order n greater than 2 cannot have a subgroupof order n – 1.

A: Given: To Prove: G cannot have a subgroup of order n-1.

Q: a. Prove that the set of numbers {1,2, 4,5, 7,8} forms an Abelian group under multiplication modulo…

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Q: A. Determine whether the given subset of the complex numbers is a subgroup of the of complex numbers…

A: Given

Q: Theorem :- Let (G,-) be a group then :- 1- (Hom(G),) is semi group with identity. 2- (A(G),) is…

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Q: Prove that The set of all complex number is a group under multiplication, and G={1,-1,i,-i}  is a…

A: Hi! You have posted multiple questions. As per norms, we will be answering only one question and as…

Q: Describe all the elements in the cyclic subgroup of generated by the 2×2 matrix     [1  1 0  1]

A: Describe the elements in the subgroup of the General linear group GLn, ℝ generated by the 2×2…

Q: 2. List the elements of the (i), i. e. cyclic subgroup generated by i of the group C* of nonzero…

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Q: give an example of a finite, non-cyclic abelian group containing a container of order 5

A: Take the abelian group G=Z5×Z5 of order 25 whose every element (except identity) is of order 5 and…

Q: Let G ={1,−1, i,−i} be the group of 4 complex numbers under multiplication.  (a) Is {1, i} a…

A: In the given question we have to determine if given set is subgroup of G or not.

Q: prove that a group of order 45 is abelian.

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Q: 3. Find all Abelian groups (up to isomorphism) of order 100. From these isomorphism classes…

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Q: 2. Let R = R – {1} be the set of all real numbers except one. Define a binary operation on R by a *…

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Q: List the elements of the , i. e. cyclic subgroup generated by i of the group C* of nonzero complex…

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Q: * Let G be the group of all non-zero complex numbers under multipilication and let G be the group of…

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Q: Prove that the group of positive rational numbers, Q+, under multiplication is not cyclic.

A: Group under addition cyclic or non cyclic

Q: Determine all subgroups of R* (nonzero reals under multiplication)of index 2.

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Q: Let Z denote the group of integers under addition. Is every subgroupof Z cyclic? Why? Describe all…

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Q: Consider the multiplicative group Z*18. List all of its elements in increasing numerical order,…

A: please see the next step for solution

Q: Prove that the set of natural numbers N form a group under the operation of multiplication.

A: The set N of all natural numbers 1, 2, 3, 4, 5... does not form a group with respect to…

Q: Q7/ Find all possible non-isomorphic groups of order 77.

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Q: 6. Embed the group Qs into the SU(2).

A: Given: Q0=e,i,j,k  e-2=e, i2=j2=k2=ijk=e, Where, e is the identity element and e commutes with the…

Q: Determine all elements of finite order in R*, the group of nonzeroreal numbers under multiplication

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Q: Prove that there is no simple group of order 525 = 3 . 52 . 7.

A: The prime factors of 525 are 3, 5 and 7. So there are proper normal subgroups of order either 3,5 or…

Q: 1.) Let G ={1,−1,i,−i} be the group of 4 complex numbers under multiplication.  (b) Is {1,−1} a…

A: Use Finite subgroup test : Let H  be  a non empty finite subset of a  group G. If H is closed under…

Q: Determine the behaviour of the set of 4 elements {1,-1,i,-i} under the ordinary multiplication of…

A: Given, The set of 4 elements are 1,-1,i,-i.It is to be shown that this set is group.

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: 3. List all elements of the cyclic subgroup of Z12 generated by 5

A: Solving

Q: Provide a two-column proof of Theorem 3: Finite Subgroup Test.

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Q: Question 5. Prove that O- subgroup of itself. (the set of positive rational numbers) under…

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Q: Prove that

A: To prove: Every non-trivial subgroup of a cyclic group has finite index.

Q: Find Aut(Z20). Use the fundamental theorem of Abelian groups to express this group as an external…

A: Find Aut(Z20) by using the fundamental theorem of Abelian groups

Q: ) Describe, up to isomorphism, all abelian groups of order 300 = 22 . 3 - 52 using the Fundamental…

A: Solution: Given : Group of order 300

Q: List the elements of the (i) , i. e. cyclic subgroup generated by i of the group C* of nonzero…

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Q: Please help me understand the following question and please explain the steps. Picture is below

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Q: 14. Prove that the set of all rational number of the form 3"6" | m,nEZ} js a group under…

A: Denote with

Q: Prove C*, the group of nonzero complex numbers under multipliation, has a cyclic subgroup of order n…

A: n th root of unity is the cyclic subgroup of order n.

Q: 12. Let "- (a + bi |a.bER, ab a 0}. Determine whether H is a subgroup of the complex numbers C with…

A: 12. Consider the group of complex numbers C with addition. Let H=a+bi|a,b∈R, ab≥0. Clearly, the…

Q: For each positive integer n, prove that C*, the group of nonzero complex numbers under…

A: Given:

Q: Show that the set R of all real numbers together with usual addition of real numbers is a group.…

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Q: Prove that if n is not prime, then {1, 2, 3,..., n-1} is not a group under multiplication mod n.

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Q: Prove that every subgroup of nilpotent group is nilpotent

A: Consider the provided question, We know that, prove that every subgroup of nilpotent group is…

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: Let p and q be distinct odd primes. Prove that Z, is not a cyclic group.

A: The Chinese remainder theorem states that if one knows the remainders of the Euclidean division of…

Q: Q1)) Prove or disprove (only two) : 1) Every subgroup of P-group is P-group.

A: The solution are next step is

Q: Let G ={1,-1, i, -i} be the group of 4 complex numbers under multiplication. (b) Is {1,–1} a…

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Q: Find all finite-dimensional complex representations of the group Z.

A: To find all finite-dimensional complex representation of the group Z.

Q: A) Prove that A5 has no subgroup of order 30

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Q: 1- Let (C,) be a group of non-zero complex number and let H = {x + iy}| x² + y² = 2}. Then (H,) is a…

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

Q: 1.) Let G={1,−1,i,−i} be the group of 4 complex numbers under multiplication.  (a) Is {1,i} a…

A: G={1,-,1,i,-i} is group under multiplication. Let H={1,i} then

Q: 2.1. Construct the group under modulo-6 addition. 2.2. Construct the group under modulo-3…

A: