Let a and b be elements of a group.  If |a| and |b| are relatively prime, show that intersects = {e}.

Q: Consider the group (Z,*) defined as a*b=a=b , then identity (Neutral) element is

A: Given that ℤ,* is a group. where * is defined as a*b=a=b. That is a-b=0. To find the neutral element…

Q: Let H be a subgroup of a group G and a, bEG. Then be aH if and only if * a-1b e H O None of these…

A: Given H is a subgroup of G. We need to find a necessary and sufficient condition for a belongs to…

Q: If a, b are elements in a group G, show that (ab)-1= b-1a-1

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Q: Let G be a group and let a,b element of G such that (a^3)b = ba. If |a| = 4 and |b| = 2, what is…

A: see below the answer

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: Let G be a cyclic group ; G=, then (c*b)^=c4* b4 for all a, c, b EG.

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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: 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: group ⟨a,b,c|b^4 = a,c^(−1) = b⟩ is abelian.

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Q: If G is a group and a1, a2,…, an shows that a1 * a2 *… * an is unique, regardless of the order in…

A: We are given that G is a group. (G satisfies all the group axioms). Suppose * is the defined binary…

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

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Q: For any group elements a and x, prove that |xax-1| = |a|.

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Q: Let G be a group and let H< G. If [G: H] = 16 and |H| = 21, then what is |G|?

A: The expression, G:H can be written as GH .

Q: Let X be a group and we then let x and y be an element of X.  Prove that (x*y)^ -1 = a^-1 * b^-1 iff…

A: Since there are some mistakes in given typed question.question may like "Let X be a group and let…

Q: Let H be a subgroup of a group G and a, b E G. Then be aH if and only if *

A: So,  a, b  belongs to H, and we have  b∈aH  Hence,  b = ah     -- for some element of H  Hence,  a-1…

Q: Let a and b be elements of a group G. Prove that G is abelian if and only if (ab)-1 = a-'b-1 %3D

A: Given: The statement is, let a and b be elements of a group G. Prove that G is abelian if and only…

Q: For any elements a and b from a group and any integer n, prove that(a-1ba)n = a-1bna.

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Q: . Let G be a group and a e G. An element be G is called a conjugate of a if there exists an element…

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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 G be a group, and let a E G.  Prove that C(a) = C(a-1).

A: Given: Let G be a group and let a∈G. then we will prove C(a)=C(a-1) If C(a) be the centralizer of a…

Q: 2. In each case determine whether the two given groups are isomorphic. Justify your answer. a) (2Z,…

A: a) Given that the groups are 2ℤ,+ and 3ℤ,+.The function is given by φ:2ℤ→3ℤ and can be defined as…

Q: Let G be a finite group and a€G s.t |a|=12.if H= find all other generators of H.

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Q: Let a and b belong to a group.   If |a| = 10 and |b| = 21, show that n = {e}

A: Consider a group G. Let a and b be elements of the group G such that a=10 and b=21. Consider the…

Q: For a group G and a fixed element a ∈ G, define the subset C(a) to be the set of all elements in G…

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Q: Let G be a group and let a be a non-identity element of G. Then |a| = 2 if and only if a = a-1.

A: Let G be a group with respect to * . Let e be the identity element,and a is non identity element.…

Q: f H and K are two subgroups of a group G, then show that for any a, b ∈ G, either Ha ∩ Kb = ∅ or Ha…

A: If H and K are two subgroups of a group G, then show that for any a, b ∈ G,either Ha ∩ Kb = ∅ or Ha…

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: If a is an element of order 8 of a group G,

A: Let G be a group. Let a be an element of order 8 of group G. That is, a8=e  where e is an identity…

Q: Let a belong to a group and a12 = e. Express the inverse of each ofthe elements a, a6, a8, and a11…

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Q: Let G be a group, and let H < G. Assume that the number of elements in H is half of the number of…

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Q: Q2: If G = R- {0} and a * b = 4ab ,show that (G,*) forms a commutative group? %3D

A: To show for the commutative group of (G, *), we verify the following properties of the commutative…

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: Suppose G is a group and r, be G so that r = b and r = b. Solve for a in terms of b.

A: Given: G is a group, and x,b∈G, so that x3=b5 and x8=b2. Formula used: Basic formula in power and…

Q: Find the order of each of the elements of the group ((Z/8Z*, * ). Is this group cyclic? Do the same…

A: To investigate the orders of the elements in the given groups

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

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Q: 6. If G is a group and a is an element of G, show that C(a) = C(a')

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Q: Q/ Let G = Z and a * b = a + b show that (G,* ) is a group or not.

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Q: Let a and b be elements in a group G. Prove that ab^(n)a^(−1) = (aba^(−1))^n for n ∈ Z.

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Q: Show that if p and q are distinct primes, then the group ℤp × ℤq is isomorphic to the cyclic group…

A: We have to show that if p and q are distinct primes, then the group Zp×Zq is isomorphic to the…

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: If G is a cyclic group of order n, prove that for every element a in G,an = e.

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

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Q: Find all the generators of the following cyclic groups: (Z/6Z,+), ((Z/5Z)*, ·), (2Z, +), ((Z/11Z)*,…

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Q: Find the solution set for each of the following with the representation of the group on the number…

A: To find - Find the solution set for each of the following with the representation of the group on…

Q: F. Let a e G where G is a group. What shall you show to prove that a= q?

A: Solution: Given G is a group and a∈G is an element. Here a-1=q

Q: Find the order of each element of the group Z/12Z under addition

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Q: The inverse of 3 in the group (Z5, o5) is

A: Definition:

Q: Let a, b be elements of a group G. Assume that a has order 5 and a³b = ba³. Prove that ab = ba.

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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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