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

Q: Prove that any group with prime order is cyclic.

A: Given, Any group with prime order. let o(G)=p (p is a prime number) we assure that G has no subgroup…

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: Prove that there is no simple group of order p2q, where p and q areodd primes and q > p.

A: Let G be a group. |G| = p2q, where p and q are odd primes and q > p. The first Sylow Theorem: A…

Q: Prove that a group of order 7is cyclic.

A: Solution:-

Q: 9. Prove that a group of order 3 must be cyclic.

A:

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 O3 is not a group

A:

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: Prove that there is no simple group of order 300 = 22 . 3 . 52.

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Q: Prove that the group G = [a, b

A: Given, the group G=a, b with the defining set of relations…

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: Prove A3 is a cyclic group

A: We know that If G be a group of prime order then G is cyclic group

Q: For any elements a and b from a group and any integer n, prove that (a21ba)n 5 a21bna.

A: For any elements a and b form a group and any integer n, We have to prove that a-1ban=a-1bna.

Q: Prove or Disprove that the Klein 4-group Va is isomorphic to Z4.

A: The statement is wrong.

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

A:

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

A: Group under addition cyclic or non cyclic

Q: Prove that, there is no simple group of order 200.

A: Solution:-

Q: For every integer n greater than 2, prove that the group U(n2-1)is not cyclic.

A:

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: Prove that in a group, (ab)^2=a^2b^2 if and only if ab=ba.

A: Proof:Let a,b ∈ G.Assume (ab)2 = a2b2 and that prove ab = ba as follows.

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: Prove that in a group, (a-1)-1 = a for all a.

A: By definition (a-1)-1=a are both elements of a-1. Since in a group each element has a unique…

Q: Prove that if (ab)' = a*b² in a group G, then ab = ba.

A: Given,ab2=a2b2To prove: ab=ba

Q: 5: (A) Prove that, every group of prime order is cyclic.

A:

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: Find order of all the elements of the Group = {1, −1, ?, −?} . The binary operator ∗ is defined as ?…

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Q: Prove that in a group, (a-1)¯' = a for all a.

A: To prove that in a group (a-1 )-1=a for all a.

Q: = Prove that, there is no simple group of order 200.

A:

Q: Prove that there is no simple group of order 210 = 2 . 3 . 5 . 7.

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Q: Prove that if (ab)2 = a?b? in a group G, then ab = ba %3D

A: A group is a set with a binary operation with following axioms satisfied. First the operation must…

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

A:

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

A:

Q: Show that a homomorphism defined on a cyclic group is completelydetermined by its action on a…

A: Consider the x is the generator of cyclic group H for xn∈H, ∅(x)=y As a result, For all members of…

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: (c) Show that Proposition 2 no longer holds without the condition that G is commutative. That is,…

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

A: The solution are next step is

Q: ) Prove that Z × Z/((2,2)) is an infinite group but is not an infinite cyclic grou

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

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Q: a) Is there any relation between the automorphism of the group and group of permutations? If exists,…

A: An automorphism of a group is the permutation of the group which preserves the property  ϕgh=ϕgϕh…

Q: 10. Prove that all finite groups of order two are isomorphic.

A: Here we use basic definitions of Group Theory .

Q: Show that each non-zero element of an integral domain , regarded as a member of the additive group…

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Q: Prove that in any group, an element and its inverse have the same order.

A: Proof:Let x be a element in a group and x−1 be its inverse.Assume o(x) = m and o(x−1) = n.It is…