7. Find order of all the elements of the Group {1,-1,1,-1}. The binary operator • is defined as a b = ab.

Q: Show that the Galois group of x 4 + x² + 1 is the same as that of x 6-1 and is of order 2.

A: We have to show that the Galois group of is the same as that  of x6 -1 and is of order 2.

Q: for each of the group Z2,Z4,Z6 INDICATE WHICH CYCLIC. FOR those cyclic list  all the generators?

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Q: Chapter 2, exercise 6.4: Prove that in a group, the products ab and ba are conjugate elements.

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Q: Fill in the blanks in the table below so that it becomes the Cayley table of a group with elements…

A: We will fill the Cayley table in a way such that it will form a group.

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: 3. Let G = Z96 Z24 Z6 O Z3. Find a direct sum of cyclic groups of prime power order that is…

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Q: Let X,Y be cyclic groups of order 2,3 respectively.  Find the orders of the elements of X×Y.

A: Let X, Y be cyclic groups of order 2,3 respectively then we have to find the   the orders of the…

Q: Let D10 = (a, b | a5 = b2 10. Assume x and y are two distinct elements of order 2 in D10. Let H =…

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Q: Let G - { (a, b) : a eR, be R\{0} }. Define on G the binary operation: (a, b) (c , d)- (a + bc , bd)…

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Q: Let Qg be the set of quaternions, defined by the matri {±1,±i,±j, ±k} where %3D Qg is an abelian…

A: For a set G to be a group under binary operation *, it must satisfy the following properties. Let…

Q: For any two elements a and b of a group G, show that G  is abelian iff…

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Q: means a cyclic group but 3+i can't turn into 3-i with a positive integer exponent.

A: Please see the answer in next step.

Q: Chapter 6, exercise 4.1: Write the product x²yx-¹y-¹x³y³ in the form x¹y¹ in the dihedral group Dn.

A: Solution:    Given equation is x2yx-1y-1x3y3,    and we have to find products in the form of xiyj in…

Q: Let X, Y be cyclic groups of order 2, 3 respectively. Find the orders of the elements of XxY.

A: Let X, Y be cyclic groups of order 2,3 respectively then we have to find the   the orders of the…

Q: Find the elements (a, b) of G of order 2. O Show that G has no element of order 3.

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Q: Let G= U(32) = {1,3,5,7,19, 1,13,15,17, 19, 21 - H = U(32)= {1,17} %3D %3D Which of the 3 Abelian…

A: We have to find  isomorphic group to G/H:

Q: Determine the Galois group of x2 +9 over R.

A: The objective of the question is to determine the Galois group of x2+9 over R.

Q: 7. Find order of all the elements of the Group = {1,–1, i, -i}. The binary operator * is defined as…

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Q: 3. Let {: ) G = | a,be Q, a² +b² # 0 26 a Determine if G is a group with respect to matrix…

A: Satisfy all four properties for proving G be a group.

Q: a+b Let G = {a € R| -1 < a < 1}. Define the binary operation on G by a b = ; Show that (G,) is a…

A: Given :  G = a∈ℝ : -1&lt;a&lt;1. And the binary operation * on G defined by  a*b = a+b1+ab ∀ a, b∈G.…

Q: (5) Prove that every group of order 3 is cyclic.

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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: Consider the group D4

A: Given: Group D4=a,b=e=1,a,a2,a3,b,ab,a2b,a3b and a=1 2 3 4, b= 2, 4 To find : The value of…

Q: Does (U(14),×14) form a cyclic group? If yes, find all generators.

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Q: Q\ Let (G,+) be a group such that G={(a,b): a,b ER}. Is ({(0,a): aER} ,+) sub group of (G,+).

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Q: Let G = (1,-1,i,-1} Prove G is a cyclic group under the multiplication operation.

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Q: Q 5 Find the inverse of the element 9 in the group (U(10), ®10). Q 6 Find the order of the element 2…

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Q: Write out the Cayley table for the dihedral group D6 = {1,r,r2,s,sr,sr2}. (In general, we apply the…

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Q: 11. Prove that the set of all 2 × 2 matrices with entries from R and de- terminant +1 is a group…

A: To show that the set of all 2×2 matrices with entries from R and determinant +1 is a group under…

Q: 2.10 Let G be the set of all 2×2 matrices ( .), where a, b ER and a² + b² +0. Show that G forms a…

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Q: Write the three-dimensional matrices that represent each of the operations of the point group D2n…

A: The dihedral group D2n is defined by the collection of symmetries of a n-sided regular polygon. So…

Q: 2.11 Let G be the set of all 2x2 matrices ), where a and bare nonzero real numbers. Show that G…

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Q: Show that U(8) is the internal Sivect product of certain two groups of U(8)

A: To show that U(8) is the internal direct product of certain two subgroups of U(8), we need to find…

Q: Develop the irreducible (2×2) matrix representations of the group of rotations (including those that…

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