6. ! Write down the relation matrix of the abelian group G specified as follows. G - {x, y, z, w | 3x + 3y + 4z – w, 6x + 4y + 13z - 7uw, 2y – 4z + 4w – 0, 9x + 9y + 13z – 4w} . Reduce this matrix using elementary integer row and column operations, and hence write G as a direct sum of cyclic groups.

Q: Compute the factor group (Z6 x Z4)/(([2]6, [2]4))

A:

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

A: Solution is given below:

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A:

Q: Determine the Galois group of x2 - 10x + 21 over Q.

A: To find- The Galois group of x2−10x+21 over Q.

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Q: 9. Classify the following groups in the sense of the Fundamental Theorem of Finitely Gener- ated…

A:

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Q: List the six elements of GL(2, Z2). Show that this group is non-Abelian by finding two elements that…

A:

Q: Determine the Galois group of x3 - 1 over Q and x3 - 2 over Q.

A: In this question, we have to determine the Galois group of x3 - 1 over Q and x3 - 2 over Q.

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

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A:

Q: (a) (Z,*) with a × b = a/b is (b) (Q, *) with a * b = ab - a - b is (c) (Q,*) with a*b = a² +6² is…

A:

Q: Chapter 2, exercise 4.11(a). For an extra challenge, you can try to prove that you only need the…

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Q: express Z10 as a product of Z5 x Z2 , verify that both groups are isomorphic.

A:

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