Prove that every subgroup of the quaternion group Qs is normal. Deter mine all the quotient groups.

Q: What is the relationship between the degree of a field extension and the size of its Galois group?

A: A field extension is a pair of fields E and F such that the operations of F are those of E…

Q: Let p be an odd prime number. Let Dp be the diehedral group of order 2p. Find |Aut (Dp)|.

A: Dp is the Diehedral group of order 2p, where p is an odd prime number.  We have to find AutDp.

Q: 11. Let K be the splitting field of x³ – 2 over Q. (Refer to Example 50.9.) a. Describe the six…

A: please see the next step for solution

Q: Let G be the group {±1, ±i, ±j, ±k} with multiplication. Show that G is isomorphic to <a, b | a2…

A: Given: G be the group ±1, ±i, ±j, ±k with multiplication.We need to show that G is isomorphic to a,…

Q: Dn Prove that is isomorphic to a subgroup of Sn

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Q: Suppose F1 and F2 are distinct reflections in a dihedral group Dn.Prove that F1F2 ≠ R0.

A: Given: F1 and F2 are distinct reflections in a dihedral group Dn. To Prove: F1F2≠R0.

Q: Let R = R\ {-1} and define the operation ♡ on R by a♡b = ab + a +b Va, be R. Show that (a) V is a…

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Q: Prove that SL,(R) is a normal subgroup of GL,(R).

A: Let G=GL(n,R) be the general linear group of degree n, that is, the group of all n×n invertible…

Q: Let R = R \ {-1} and define the operation ♡ on R by a♡b = ab + a + b Va, b E R. Show that (a) ♡ is a…

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Q: Prove that GL(2,R) is not an abelian group

A: Solution is given below:

Q: Prove or disprove that C* has a subgroup isomorphic to Z2 ⨁ Z2.

A: Given: Set, Z2⊕Z2We need to prove or disprove that C* has a subgroup isomorphic to Z2⊕Z2.

Q: Show that Z12 is not isomorphic to Z2 ⊕ Z6. Recall that ℤ? denotes the abelian cyclic group of order…

A: It has to be proved that Z12 is not isomorphic to Z2 ⊕ Z6. Isomorphism is defined as one-to-one…

Q: 1. Consider H= (i) and K= (-1) the subgroups of Q, the quaternion group. 1) Explain why H is normal…

A: Basic theory regarding normal subgroups from algebra will be used

Q: Let m ≥ 3. Prove that the symmetric group on m letters is not abelian.

A: We have to show that the permutation of m letters, m≥3 that is Sm is not abelian. For proving the…

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

A:

Q: 1.) Let G₁ = £1₁ 1₁ i -i) be group of 4 Complex numbers under mutiplication. a) is £1, i} a subgroup…

A: Given group under multiplicationG = 1, -1,  i , -i  A subset H of G will be subgroup of G if H…

Q: Draw a Venn diagram with three overlapping circles for the three given sets. Label the contents of…

A: Venn Diagram for the problem involves the use of 3 overlapping circles for the sets So Venn diagram…

Q: Prove that E(n) = {(A, ¤) : A e O(n) and E R"} is a group. %3D

A: Consider the given: E(n)={(A,x)} where A∈O(n)and x∈ℝn

Q: Let G be a group and let r, y e G such that ya = r-ly. Use the Principle of Mathematical Induction…

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Q: 1) Prove that G = {a + bv2 \a,b € Q and a, b # 0} is a subgroup of (R – (0},.)

A: Subgroup

Q: 2. In this exercise, we will construct a new finite group of order 8 using the quater- nions. The…

A: (a)  Recall  that the Quaternion group is  the group which consists   of the  eight quaternions .…

Q: describe all the elements in the cycle subgroup GL(2,R) generated by the given 2 x 2 matrix [0 -1]…

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Q: 3. Define Lie group.

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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: 2. Let G = Z50 and consider it's subgroup H = (5). Find all coset representatives of 3+ H.

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Q: Prove that every subgroup of the quaternion group Q8 is normal. Deter- mine all the quotient groups.

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Q: and m' be two R-modules. e set Hom, CM, Mm) is an group under pointwise of morphism

A:

Q: Use Lutz-Nagell's theorem and reduction mod p theorem to show that the torsion group of E : y² = x³…

A: Given: Use Lutz-Nagell's theorem and reduction mod p theorem. To proof: Torsion group of E:y2=x3+3…

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: Write Q(sqrt2) for the set described in exercise 3.1 (iii). given a non zero element a + bsqrt(2) of…

A: It refers to the set Q(2), which is the set of all numbers of the form a+b2, where a and b are…

Q: :R*, multiplication> is

A: A torsion group (also called periodic group) is a group in which every element has finite order. 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: 4. The table below is known as the Ouaternion Group. Use the table below to evaluate the following:…

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Q: Let (G,*) be a group. Show that (G,*) is abelian iff (x * y)² = x² * y² for all x, y E G.

A: If a group G is abelian, then for any two elements x and y,    (x*y) = (y*x)   now associative…

Q: Let C be a group and let a be an element of C Prove that = .   Please be clear with theorems ana…

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Q: Let G be a group and let a, be G such that la = n and 6| = m. Suppose (a) n (b) = (ea). Prove that…

A: According to the given information, let G be a group.

Q: Problem 3. Prove that if m ± 1n, then the symmetric groups S and S. are not isomorphic.

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Q: Determine the galois group

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Q: 8. Let G = U = {z e C| |z| = 1} be the circle group. Then X = C, the set of complex numbers, is a…

A: Given: G=U=z∈ℂ|z=1 is the circle group. X=ℂ, the set of complex numbers, is a G-set with group…

Q: Let F and H be subgroups of group G and let FCH. Prove (G : F) = (G : H)(H : F).

A: It is given that F and H are subgroups of G and F⊂H.

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: Show that A4 is nolmal by finding a homomorphism whose kernel is A4.

A: If (alternating group) An is the normal subgroup of Sn, then by first isomorphism theorem implies…