Show that the translations of Rn form a group.

Q: Prove that any group with three elements must be isomorphic to Z3.

A: Let (G,*)={e,a,b}, be any three element group ,where e is identity.  Therefore we must have…

Q: If g and h are elements from a group, prove that ΦgΦh = Φgh

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

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

A: We have given a map ,  ϕ : A → B , where A = ℝ , +   ,  B = ℝ* , · such that  , ϕx = 2x We know that…

Q: Prove O3 is not a group

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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: · In a group, prove that (ab) = b-'a-|

A: As you asking for question number 7 , I solve for you.

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

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Q: Prove that a group of order 12 must have an element of order 2.

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

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

A: Given the order of the group is 3, we have to prove this is a cyclic group.

Q: Suppose G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G,…

A: Let G is a group and Z (G) and lnn (G) are the centers and groups of internal deformations of G

Q: Let G be a group and let g, h ∈ G. Show that | gh | = | hg |. Remember that | a | denotes the order…

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

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Q: a The group is isomorphic to what familiar group? What if Z is replaced by R?

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Q: prove that any group R=3 must beperiedio

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Q: Assume that the equation yxz = e holds in a group. Then *

A: If a is the inverse of b, then it must be that b is the inverse of a.

Q: Can you prove that a set is a group, without having an operation? for example can you prove this set…

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

A: Solution:-

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: Show that ( Z,,+,) is a cyclic group generated by 3.

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Q: Prove that G = {a+b√2: a, b € Q and a and b are not both zero} is a subgroup of R* under the group…

A: Given- G=a+b2: a,b∈ℚ and a and b are not both zero To prove- G is a subgroup of ℝ* under the group…

Q: Prove that if a is the only element of order 2 in a group, then a lies inthe center of the group.

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Q: If H and K are subgroups of a group G, prove that ANB is a subgroup of G.

A: GIVEN if H and K are the subgroup of a G, prove that  A∩B  is a subgroup of G

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

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Q: If A is a group and B is a subgroup of A. Prove that the right cosets of B partitions A

A: Given : A be any group and B be any subgroup of A. To prove : The right cosets of B partitions A.

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: State the first isomorphism theorem for groups and use it to show that the groups/mz and Zm are…

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Q: 5. D, =

A: First we have to show that the dihedral group is  D_2n is solvable for n>=1

Q: Show if the shown group is cyclic or not. If cyclic, provide its generator/s for H H = ({3* : k E…

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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: Let H and K be subgroups of a group G. If |H| = 63 and |K| = 45,prove that H ⋂ K is Abelian.

A: Given: The H and K are subgroups of a group G. If |H| = 63 and |K| = 45 To prove that H ⋂ K is…

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: 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 x be in a group G. If x' - e and x* - e , prove that x - e and x' = e

A: Let G be a group and x∈G.Given: x2≠e and x6=e , where e is the identity element.To Prove: x4≠e and…

Q: Let H be a subgroup of a group G with a, b ϵ G. Prove that aH= bH if and only if a ϵ bH.

A: For the converse, assume a-1b∈H, we want to show aH=bH Let a-1b=h for h∈H. Suppose x∈aH. Let x=ah1…

Q: Let G = {a + b/2|a, b € Z}. Show that G is a group under ordinary addition.

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

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Q: Let G be an abelian group, then (acba)(abc)¯1 is

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Q: Let G be a group, a E G. Prove that a=a + a < 2

A: Concept:

Q: If A is an abelian group with A <G and B is any subgroup of G, prove that ANB < AB.

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Q: . Prove that the group Zm × Zn is cyclic and isomorphic to Zmn if and only if (m, n) = 1.

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Q: G1 = Z÷ and G2 = Z, *

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Q: G be the external direct product of groups G, G2.. H, = {4,e2.e*, e..e,x, e G,}

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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…

Q: Verify that (ℤ, ⨀) is an infinite group, where ℤ is the set of integers and the binary operator ⨀ is…

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Q: Show that a group of order 77 is cyclic.

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