Suppose c is a conjugacy class in a с group G such that IC is finite but | c/#1. Then there exists an cement of in G that does g not commute with element of C. Show that hypothesis of any finiteness of IC | is necessary for this.

Q: Show that the group of positive rational numbers under multiplicationis not cyclic. Why does this…

A: To Show: A group of positive rational numbers under multiplication is not cyclic and why does this…

Q: Suppose that g and h induce the same inner automorphism of a group G. Prove that h-'g E Z(G).

A: Given: g , h induce the same inner automorphism of a group G. Prove that: h-1g∈ZG.

Q: 4. Suppose (G₁, *) and (G₂, 0) are groups, let i and J denote the respective identities of G₁ and…

A: and are group. be a group isomorphism.We have to prove that if  has finite order the  has finite…

Q: Prove that every group of order 330 is not simple.

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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: Show that U5 andZ4 are isomorphic groups?

A: U(5)= {1,2,3,4}, <2> = {2, 22 = 4, 23 = 8, 24 = 1} = U(5) Therefore, U(5) is a cyclic group of…

Q: Prove O3 is not a group

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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: Prove that there is no simple group of order n, where 201 ≤n ≤ 235 and n is not prime.

A: We have to prove that there is no simple group of order n, where 201≤n≤235 and n is not prime. First…

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

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Q: 5. Let G be a group of order p'q, where p, q are prime numbers, and q # 1(modp), p² # 1(modq). Prove…

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Q: Consider the set of integers with the operation * given by a b=a -b. Which of the following are…

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Q: Give an example of a non-Abelian group that has exactly threeelementsof finite order.

A: Let G=Z3⊕Z be an additive group where e=0,0. Let x∈G then it can be observed that x=a,b where a∈Z3,…

Q: ab 2. Show that (Z, *) is an abelian (commutative) group, where' is defined as a*b = and Z is the…

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Q: (d) Define * on Q by a * b = ab. Determine whether the binary operation * gives a group on a given…

A: Note: Hi! Thank you for the question as per the honor code, we’ll answer the first question since…

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

A: We need to prove that there is no simple group of order p2q where p and q are odd primes and q>p.…

Q: Show that the order of the group of automorphism of a finite group of order n is a divisor of n!.

A: we have to show that the order of the group of automorphism of a finite group of order n is a…

Q: B Suppose (G, *) is group of order > 1. Which statement best describes problem? Cannot have zero,…

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Q: Without using the Generalized Cayley Theorem or its corollaries,prove that there is no simple group…

A: A simple group is a group with no non-trivial proper normal subgroups. We need to show that there is…

Q: Let G = Z4 x Z6. Compute the factor groups G/((2, 3)) and G/((3, 3)). (In each case, write the…

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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: What are all abelian groups of order 81 up to isomorphism? Are there any Theorems I can use to…

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Q: Consider a group G with operation * and identity element e. Show that for any element a in G, there…

A: In any group G, we define a group as a set G equipped with an operation * that combines any two…

Q: G is a finite group and f be an automonphism of G satisfying the conditian f(x)= x → x=e for xEG.…

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Q: Show that a group G order pqr for distinct primes p ,q and r ,is solvable

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Q: Prove that there are exactly five groups with eight elements, up to isomorphism.

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Q: Show that for an abelian group, the only inner automorphism is the identity mapping, whereas for…

A: We have to show that for an abelian group, the only inner automorphism is the identity mapping,…

Q: Show that the quotient group Q/Z is isomorphic to the direct sum of prufer group

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Q: Show that if G and H are isomorphic group, then G commutative implies H is commutative also.

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Q: Construct the group Z4 building an isomorphism. Z8/ and determine a group that it is isomorphic to…

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Q: Every quotient group of an abelian group G  is abelian but converse is not true.

A: Every quotient group of an abelian group G is abelian but converse is not true.