1.(a) (i) Let G = R-{0} × R. Prove that (G, *) is a group where * is defined by(a, b) * (c, d) = (ac, bc + d).(ii) Let H be the multiplicative group of non-singular 2 x 2 matrices with rational entries.Determine the orders of the following elements of G.112(b) Let o = (1 6 9)( 2 4)(3 5 7) be an element of S9.(i) Find the order and parity of o.(ii) Hence, find o35.(iii) Write o as a product of transpositions.

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(a) (i) Let G = R-{0} × R. Prove that (G, *) is a group where is defined by (a, b) * (c, d) = (ac, bc + d).

(a) (i) Let G = R-{0} × R. Prove that (G, ) is a group where is defined by (a, b) (c, d) = (ac, bc + d).

Elements Of Modern Algebra

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ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

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Chapter3: Groups

Section3.5: Isomorphisms

Problem 8E: Exercises 8. Find an isomorphism from the group in Example of this section to the multiplicative...

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Question

1.
(a) (i) Let G = R-{0} × R. Prove that (G, *) is a group where * is defined by
(a, b) * (c, d) = (ac, bc + d).
(ii) Let H be the multiplicative group of non-singular 2 x 2 matrices with rational entries.
Determine the orders of the following elements of G.
1
1
2
(b) Let o = (1 6 9)( 2 4)(3 5 7) be an element of S9.
(i) Find the order and parity of o.
(ii) Hence, find o35.
(iii) Write o as a product of transpositions.

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