5. Let G be a group and let a € G. An element b E G is called a conjugate of a if there exists an element x E G such that b = xax-1. Show that any conjugate of a has the same order as 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: Prove that if x is a group element with infinite order, then x^m is not equal to x^n when m is not…

A:

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

A:

Q: 6. Show that for any two elements x, y of any group G, o(xy) = o(yx). %3D

A: Fact 1:In a group G, if x∈G such that xn = e, then O(x)|n (e→ identity element.)i.e. order of x…

Q: prove that the group G=[a b] with defining set of relations a^3=e, b^7=e, a^-1ba=b^8 , is a cyclic…

A: We need to prove that , group G = a , b  with defining sets of relations a3 = e   ,  b7 = e  also…

Q: Define ∗ on ℚ+ by a ∗ b =ab/2 . Show that ⟨ℚ+,∗⟩ is a group.

A:

Q: If x is an element of a cyclic group of order 15 and exactly two of x3, x5, and x9 are equal,…

A: Given: The order of group is 15

Q: (H,*) is called a of (G,*) if (H,*) is a group.

A:

Q: Suppose T: G-> H is group isomorphism and S:H-> K is also a group isomorphism. SHow the…

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Q: Is the set Z a group under the operation a * b = a + b – 1? Justify your answer.

A:

Q: Let G be a group and suppose that a * b * c = e. Show that b * c *a = e.

A:

Q: Let G be the group with presentation (x,y: 1² = 1, r² = y²₁ xy = y ¹). Decide how many elements are…

A: Consider the given group G with presentation, x,y:x4=1, x2=y2, xy=yx-1 it can be observed that it is…

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

A:

Q: Let C be a group with |C| = 44. Prove that C must contain an element of order 2.

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Q: G let then. [b, a]= be an group and Ta %3D

A: Given that G is a group and also a,b,c∈G. To prove that b,a= a,b-1 Since G is a group, it satisfies…

Q: Let a and b be elements of a group.  If |a| and |b| are relatively prime, show that intersects =…

A: Let m and n be the order of the elements a and b of a group G. Given that the orders of a and b are…

Q: 1. Assume (X, o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) |x € X,y E Y} and define the…

A:

Q: 10, Let (G, *) be a group and a, b, c E G. (a) Prove if a *b = a * c, then 6 = C (b) Prove if b * a…

A:

Q: 2. Let G be a group. Pro-

A: Let G be a group .

Q: 1- Prove that if (Q - {0},) is a group, and H = 1+2n 1+2m 9 n, m e Z} is a subset of Q-{0}, then…

A: We need to prove that H=1+2n1+2m∋n,m∈Z is a subgroup of Q-0 A subset W of a group V is said to be a…

Q: Let C be a group with |C| = 44. Prove that Cmust contain an element of order 2.

A:

Q: Let G be a group (written additively), S a nonempty set and M (S, G) the set of all functions from S…

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Q: Can you write a group homomorphism as φ (gh) as φ(hg)? Are they the same thing?

A: The given homomorphism ϕgh, ϕhg The objective is to find whether the ϕgh,ϕhg are same.

Q: Q2) If G = Z24 Group a) Is a G=Z24 cyclic? Why b) Find all subgroups of G = Z24 c) Find U,(24)

A: Given that G=ℤ24. a) Then G is generated by the element 1. That is, 1=1,2,3...,22,23,0=ℤ24.…

Q: 6. Prove that if G is a group of order 231 and H€ Syl₁(G), then H≤ Z(G). me

A:

Q: 3. Show that G = {a+bv2 |a,b e Q} is a group under the usual addition.

A: 3. Let G is said to be group under binary operation (.). It should satisfy the following properties.…

Q: Suppose H and K are subgroups of a group G.  If |H|=12 and |K| = 35, find |H intersected with K|.…

A:

Q: Assume (X,o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) | æ € X, y E Y} and define the…

A: The given question is related to group theory. Given: X , ∘ and Y , ∙ are groups.  Let X × Y = x,y |…

Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({a +bv2 : a,…

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Q: 4. Prove that the set H = nEZ is a cyclic subgroup of the group GL(2, R).

A:

Q: Is the set Z a group under the operation a * b = a – b + ab? Justify your answer.

A: Check the associative property. Take a = 2, b = 3 and c =4. (a*b)*c = (2*3)*4              =…

Q: Show that the set S = (1, i, - 1, -0 is an abelian group with respect to the multiplication

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Q: 2. Show that in a group if x has inverse y and y and a right an inverse r, then y and r are the same…

A: We need to show that in group if x has an inverse y and a right inverse r, then y and r the same…

Q: any two elements x and y in a group G, there is a unique element z in G such that y = xz.

A:

Q: Use the fact that a group with order 15 must be cyclic to prove: if a group G has order 60, then the…

A:

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: Suppose that G = (a), a e, and a³ = e. Construct a Cayley table for the group (G,.).

A:

Q: a group and H, K be Subgroups of NG (H) = NGCH) Relate H and K? let G be G Such that %3D

A: Given: Let G be the group and H, K be the subgroups of G such that NG(H)=NG(K)

Q: Decide whether (Z, -) forms a group where : Z xZ Z (a) is the usual operation of subtraction, i.e.…

A: NOTE: According to guideline answer of first question can be given, for other please ask in a…

Q: Let c and of d be elements of group G such that the order of c is 5 and the order of d is 3 respec-…

A:

Q: Consider the set of permutations V = {(1), (1 2) (3 4), (1 3) (2 4), (1 4) (2 3)}. Determine whether…

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Q: Let G be a group, and assume that a and b are two elements of order 2 in G. If ab = ba, then what…

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Q: 64

A: Under the given conditions, to show that the cyclic groups generated by a and b have only common…

Q: 7. Prove that if G is a group of order 1045 and H€ Syl₁9 (G), K € Syl (G), then KG and HC Z(G).

A: 7) Let G be a group of order 1045 and H∈Syl19(G) , K∈Syl11(G).  To show: K⊲G and H⊆Z(G). As per…

Q: Suppose H and K are subgroups of a group G. If |H| = 12 and |K| = 35, find |H N K|. Generalize. %3D

A: Given that H and K are subgroups of a group G. Also, the order of H is H=12 and the order of K is…

Q: The set numbers Q and R under addition is a cyclic group. True or False then why

A: Solution

Q: Let G = Zp × Zp. Is this group cyclic? As you know any cyclic group can be generated by one element.…

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Q: Suppose now that we have two groups (X,o) and (Y, *). We are familiar with the Cartesian product X x…

A: Let X,◊ and Y,* are two groups. The Cartesian product of X and Y defined by X×Y=x,y | x∈X and  y∈Y.…