. Let G be a group and a e G. An element be G is called a conjugate of a if there exists an element a € G such that b = xax-. Show that if b is a conjugate of a then o(b) = o(a).

Q: a. Show that (Q\{0}, * ) is an abelian (commutative) group where * is defined as a ·b a * b = .

A: To show this we have to show that this holds closure, associative, identity, inverse and commutative…

Q: For any two elements a and b of a group G is abelian iff (ab)² = a²b². , show that G

A: For any two elements a and b of a group GClaim : G is abelian iff (ab)2 =a2b2 .

Q: Let G be a group and let a, b E G. (a) Prove that o(ab) = o(ba). (Note that we are not assuming that…

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Q: Show if the given set is a group defined by the given operation. Check all properties. If so, check…

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Q: If a is a group element, prove that every element in cl(a) has thesame order as a.

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Q: For any two elements a and b of a group G, show that G  is abelian iff…

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Q: Give an example to show that there exists a group with elements a and b such that la] = 0, \b| = ∞,…

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Q: Let a be a group element of even order. Prove that a2 is not in cl(a).

A: Let G be a group and a is an element of even order in the group. Prove that a2 is not in Cla.…

Q: Given that A and B is a group. Find out if : A→B is a homomorphism. If it is a homomorphism, also…

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Q: let x and g be elements of a group G. show that x and gxg-1 have the same order. now prove that xy…

A: Hence, proved.Explanation:Step 1:To prove that x and gxg-1  have the same order for any elements x…

Q: Let P₂(x) = {ax² + b | a, b e R and a #0}. If p(x) = ax² +b and g(x) = cx² + d are elements of…

A: In the given problem we have to show the the set p2(x) is a group with respect to the binary…

Q: Let G = U(32) and H = {1, 15}. The group GIH is isomorphic to one of Zg, Z, O Z, or Z, O Z, O Z,.…

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Q: If G is a group and g E G, show that the number of conjugates of g E G is [G : CG(g)]

A: Given G be a group and g∈G be an element. Let Bg be the set of all conjugate elements of g∈G.…

Q: 1. Let (G, +) and (H,*) be groups. The direct product of G and H is the set G × H equipped with…

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Q: Let (G, ★) and (G,o) be groups, with respective identity elements e and ē. Suppose there is a…

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Q: 3 be group homomorphisms. Prove th. = ker(ø) C ker(ø o $).

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Q: If (G, ) is a group with a = a for all a in G then G is %D abelian

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Q: Let G = {a +b 2 ∈ ℝ │ a, b ∈ ℚ }. Prove that the nonzero elements of G form a group under…

A: Note : In the question, it has to be a + b2 ∈ ℝ instead of a + b 2 ∈ ℝ. So, we are solving the…

Q: Let a be a group element of even order. Prove that a2 is not in cl(a).

A: Given that a be a group element of even order. We have to prove that a2 is not in cla.

Q: a) Check whether G = (3n, ·) is a group, where n ∈ Z. b) Let p1 (x) = x3 + 1 and p2 (x) = x4 + x.…

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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: Problem 13. Integrate f(x, y, z) = 12xz over the region in the first octant (x, y, z ≥0) above the…

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Q: (G, .> be a group such that a.a = e for all a E G. Show that G is an abelian grou 2. Let

A: We have to solve given problem:

Q: 1. Let (G. *) be a group and a E G. Suppose that a *a = a · Prove or disprove that a must be the…

A: We have to solve given problem:

Q: Given that A and B is a group. Find out if þ: A→B is a homomorphism. If it is a homomorphism, also…

A: Group homomorphism is nothing but a function defined between two groups. The function must be closed…

Q: Suppose that we have an action of a group G on a set A. For any g E G and a E A, prove that the…

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Q: Let n be a positive integer, and let nZ = {nm|m ∈ Z}. (a) Show that ⟨nZ, +⟩ is a group. (b) Show…

A: It is given thatnZ=n m |m ∈ ZWe have to show (nZ,+) is a groupnZ=n m |m ∈ ZClosure property :- Let…

Q: Let G be a group with operation * and e be the identity element of G. Show that for any element a in…

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Q: a b (88) 0 d Let G be the set of all 2 × 2 matrices under matrix multiplication. Is G abelian? where…

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Q: Let G be the set (te, ±a, ±b, ±c) where [ 1 10 Show that G form a Group under multiplication.

A: Group: If a set (finite or infinite) together with a binary operation satisfies the four basic…

Q: Prove that a group of order pq is abelian where p = 9 and q= 5.

A: We have to prove that a group of order pq is abelian where p = 9 and q= 5.

Q: 3. Let G be a group with identity e. Show that if a ★a=e for all a in G, then G is Abelian.

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Q: Show that the multiplicative group Z is isomorphic to the group Z2 X Z2 8,

A: We know that if two groups are isomorphic than they have same number of elements i.e. their…

Q: Let G1 and G2 be groups. Prove that G1 × G2 is isomorphic to G2 × G1.

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Q: Let (G,*) be a group such that a² = e for all a E G. Show that G is commutative.

A: A detailed solution is given below.

Q: Let G be a group. Prove that (ab)1= a"b-1 for all a and b in G if and only if G is abelian

A: First, consider that the group is abelian. So here first compute (ab)-1 for a and b belongs to G,…

Q: Let S = {x €R | x + 3}. Define * on S by a * b = 12 - 3a - 3b + ab Prove that (S, *) is a group.

A: The set G with binary operation * is said to form a group if it satisfies the following properties.…