Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 187(a)(c)(b)Figure 10.10.(b) 〈x, y + w〉 = 〈x, y〉 + 〈x, w〉.(c) 〈αx, y〉 = 〈x, αy〉 = α〈x, y〉.(d) 〈x, x〉 ≥ 0 with equality exactly when x = 0.(e) If 〈x, y〉 = 0 for all x in R n , then y = 0.6. Verify thatis a group.E(n) = {(A, x) : A ∈ O(n) and x ∈ R n }7. Prove that {(2, 1), (1, 1)} and {(12, 5), (7, 3)} are bases for the same lattice.8. Let G be a subgroup of E(2) and suppose that T is the translation subgroupof G. Prove that the point group of G is isomorphic to G/T .9. Let A ∈ SL 2 (R) and suppose that the vectors x and y form two sides of aparallelogram in R 2 . Prove that the area of this parallelogram is the sameas the area of the parallelogram with sides Ax and Ay.10. Prove that SO(n) is a normal subgroup of O(n).11. Show that any isometry f in R n is a one-to-one map.12. Show that an element in E(2) of the form (A, x), where x ≠ 0, has infiniteorder.13. Prove or disprove: There exists an infinite abelian subgroup of O(n).14. Let x = (x 1 , x 2 ) be a point on the unit circle in R 2 ; that is, x 2 1 + x 2 2 = 1. IfA ∈ O(2), show that Ax is also a point on the unit circle.15. Let G be a group with a subgroup H (not necessarily normal) and a normalsubgroup N. Then G is a semidirect product of N by H if• H ∩ N = {id};• HN = G.Show that each of the following is true.(a) S 3 is the semidirect product of A 3 by H = {(1), (12)}.