Q6/ Answer only two: a) Is the group of order 30 simple ,explain. Loroun G such that (G/H) = 2, prove that H is normal in G.

Q: Provide an example of the following and explain why it works: 1.) A Galois extension of Q with…

A: Introduction: The Galois group of a certain kind of field extension is a particular group connected…

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Q: Prove that the mapping from R under addition to SL(2,R) that takes x to  [ cos x     sin x  -sin x…

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Q: Exercises 3(a) Prove that the groups $ and D6 are isomophic ; (e) Preve that the additive and the…

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Q: G be defined by f(r) = x1. Prove that f is operation-preserving if 6*. Let G be a group and f: G and…

A: To prove that the given function f is a homomorphism (operation preserving) if and only if G is…

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

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Q: Here is the question I'm needing help with.  Prove C*, the group of nonzero complex numbers under…

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Q: Let G be a group of odd order. Show that for all a E G there exists b E G such that a = b?.

A: Consider the given information, Let G be a group of odd order then, |G|=2k+1 where k belongs to…

Q: This exercise refers to the triangular symmetry group discussed in this section.Which element is the…

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Q: Let Z be the group of the integes. oddition. Let f:232120e tre homonorphism fiam 2oZg. defined with…

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Q: Find all distinct subgroups of the quaternion group Qs, where Q8 = {+1,±i,±j, £k} Deduce that all…

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Q: KE Syl-(G). Prove that (a). HG and KG. (U). G has a cyclic subgroup of order 77. Syl(G),

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Q: find Aut(Z30).  Use the FUndamental Theorem of Abelian Groups to express this group as an external…

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Q: Show that 1 y? ry under a Lie group corresponding to the symmetry generator X = r² a + ry

A: Given,

Q: be a Galois extension with Galois group G. Then F = KG.

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Q: chow that An s a Group with respect, to Cooplesition of functjon.

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Q: Exercise 1) Consider the group (S3, 0) and H= {e, fs). Prove that HS3. 2) Consider the group (Z.)…

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

Q: (1) Z/12Z

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Q: Q3:(A) Prove that every group of order 15 is decomposable and normal. (B) Show that (H,.) is a…

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Q: Q3: (A) Prove that 1. There is no simple group of order 200.

A: Simple group of order 200

Q: G = (R, +), H = {a+bv2: a,b € Z}

A: Given G = (R, +), H = {a+b√2 : a, b∈Z}. We check whether H is a subgroup of G.

Q: 46. Determine whether (Z, - {0},6 ) is it a group or not? Explain your answer?

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Q: Let Dg be the Dihedral group of order 8. Prove that Aut(D8) = D8.

A: We have to solve given problem:

Q: Using the Theorem of Lagrange, prove that a group G of order 9 is abelian

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Q: Prove: (R+) (Q++) (Rx) ) X) all are non-cyclic group ?

A: Cyclic Group: A group G is called cyclic if there is an element a in G such that  G=a=an| n∈Z, where…

Q: Determine the order of (Z ⨁ Z)/<(2, 2)>. Is the group cyclic?

A: Given, the group                 We have to find the order of the group and also check, This is a…

Q: (2) (Z x Z)/(6Z × 14Z)

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Q: Use the three Sylow Theorems to prove that no group of order 45 is simple.

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Q: ii) Find the structure of its Galois group, G.

A: To Determine :- The structure of its Galois group, G.

Q: (7) Define GL2 (R) to be the group of invertible 2 x 2 matric manifold, cc this group has the…

A: Define GL2R  to be the group of invertible 2×2 matrices.  To prove that this group has the structure…

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Q: Q2 : Find the left regular representation of the group Z5 and express the group element in the…

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Q: Q4: Consider the two group (Z, +) and (R- {0}, ), defined as follow if n EZ, f(n) ={1 if nE Z, %3D…

A: Homomorphism proof : Note Ze denotes even integers and Zo denotes odd integers.  So f(n) = 1 if n is…

Q: Let G be the group of symmetries of a circle and R be a rotation of the circle of V2 degrees. What…

A: Given:

Q: 4. Construct a 2-dimensional CW-complex whose fundamental group is Z x Z/2 (and prove it).

A: Please check the detailed sol" in next step

Q: Exercise 1) Consider the group (S3, 0) and H= {e, f3}. Prove that HS3. 2) Consider the group (Z¸ +)…

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Q: 2) Determine whether or not the groups Z10 × Z4 and Z, × Z20 are isomorphic. Justify your answer.

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Q: a dihedral group Dyz (a,bia'- b°- (ab)`=4>. Consider i) Discuss all the possible Commutators of Dy…

A: Let us first find the conjugacy classes for D4.

Q: 9. Show that the two groups (R',+) and (R'- {0}, -) are not isomorphic. | 10. Prove that all finite…

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Q: Q2// Let Hi family of subgroups of (G, *). Prove that the intersection of Hi is also * .subgroup

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Q: In the set of real numbers IR there is an operation defined as: I x y= Vr³+y³ %3D prove that (IR, ×)…

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Q: 15*. Find an explicit epimorphism from Z24 onto a group of order 6. (In your work, identify the…

A: To construct a homomorphism from Z24 , which is onto a group of order 6.

Q: A) Prove that A5 has no subgroup of order 30

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Q: Let (IR, +) be a group of real numbers under addition and (R+,-) be the group of positive real…

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Q: sume that K and F are finite fields. Prove that the group Gal(KL/L) is finite cyclic.

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Q: Find the Galois group of the polynomial r-1.

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Q: (4) Find the Galois group of the polynomial r + 1.

A: Since you have asked multiple question, we will solve any one question for you. If you want any…