Show that the set R of all real numbers together with usual addition of real numbers is a group. b)Show that the set R is a semi group under multiplication of real numbers c)Prove that the identity element of a group is unique

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

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Q: Give all the possible elementary divisors of a group of order 40.

A: 40=23×5 So, the possible elementary divisors of the group are 2,2,2,5,2,4,5 and 8,5

Q: Is the set of integers a commutative group under the operation of addition? Yes; it satisfies the…

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Q: Prove that the set of all orientation-preserving isometries of R³ is a group.

A: We know that the identity transformation i is an orientation-preserving and also an isometrics as…

Q: (a) Explain why it is impossible for any set of (real or complex) numbers which contains both 0 and…

A: To solve the given problem, we use the defination of group.

Q: Prove that for a fixed value of n, the set Un of all nth roots of 1 forms a group with respect to…

A: Fix an n∈ℕ. Let U=u∈ℂ | un=1 .  Note that U⊂ℂ* and ℂ* is a group under multiplication. Let u,v∈U…

Q: In the group of integers Z with addition, cl(5) the conjugation class of 5 is equal to:

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Q: 3. Prove that the groups (R.) and (R,+) are not isomorphic, where (R* R{0},-) is the multiplicative…

A: We have to show that the groups ℝ, + and ℝ*, · are not isomorphic. Here, the group ℝ*, · does not…

Q: Construct an element of multiplicative group of the finite field elements.

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Q: 5. Define the right regular action of a group G on itself. Show it is a group action. Is it…

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Q: Let m be a positive integer. If m is not a prime, prove that the set {1,2,..., m – 1} is not a group…

A: We show that it doesn't satisfy clousre property.

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

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Q: Prove that Q+, the group of positive rational numbers under multiplication,is isomorphic to a proper…

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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: 4. Let R* be the set of real numbers except 0. Define on R by letting a*b= ab. Decide whether R with…

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Q: Let F denote the set of first 8 fibanacci number .Then convert F into a non abelian group by showing…

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Q: Prove that there is no simple group of order 216 = 23 .33.

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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 the order of group G =8, show that G must have an element of order 2.

A: Let G be a group and O(G)=8 Also Let a∈G be an arbitrary element other than identity. Then the…

Q: Let G be a finite group of order p  where p is prime. Prove that G is cyclic.

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Q: Prove that the additive group L is isomorphic to the multiplicative group of nonzero elements in

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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: 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: 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: Prove that the 2nd smallest non-abelian simple group is of order 168.

A: Introduction- An abelian group, also known as a commutative group, is a group in abstract algebra…

Q: 9. Let (G,*) be a finite group of order pq, where p and q are prime numbers. Prove that any non…

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Q: Prove that a factor group of a cyclic group is cyclic

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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: Suppose c is a conjugacy class in a с group G such that IC is finite but | c/#1. Then there exists…

A: Please find the answer in next step

Q: Show that any finite subgroup of the multiplicative group of a fieldis cyclic

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Q: Theorem: Any non-commutative group has at least six elements

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

A: Given:

Q: Characterize those integers n such that the only Abelian groups oforder n are cyclic.

A: According to the question,

Q: Show that the set S of all symmetries of an equilateral triangle form а. a group under composition…

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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: List all abelian groups (up to isomorphism) of order 600

A: To List all abelian groups (up to isomorphism) of order 600

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

A: Solution

Q: (a) Prove that any subgroup of Sn contain either even permutations only or equal number of even and…

A: As per guidelines we are allowed to solved one question at a time so i am solving first one please…

Q: Prove that the bijective functions of a set A in it, with the operation composition form a group.

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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: Exercise 3: Prove that every element of a finite group is of a finite order.

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Q: Let (IR, +) be a group of real numbers under addition and (R+,-) be the group of positive real…

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Q: Prove that the set of all rational numbers of the form m/n, where m, n E Z and n is square- free, is…

A: We prove it using a One-step test for subgroups.

Q: Prove that there is no simple group of order 315 = 32 . 5 . 7.

A: Prove that there is no simple group of order 315=32·5·7.

Q: Give an example of a subset of a ring that is a subgroup underaddition but not a subring.

A: Give an example of a subset of a ring that is a subgroup under addition but not a subring.