prob 51 42 CHAPTER 3. GROUPS 43. Prove or disprove: SL2(Z), the set of 2x2 matri

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prob 51

42 CHAPTER 3. GROUPS 43. Prove or disprove: SL2(Z), the set of 2x2 matrices with integer entries and determinant one, is a subgroup of SL2(R) 44. List the subgroups of the quaternion group, Qs 45. Prove that the intersection of two subgroups of a group G is also a subgroup of G. 46. Prove or disprove: If H and K are subgroups of a group G, then HU K is a subgroup of G 47. Prove or disprove: If H and K are subgroups of a group G, then HK(hk he es H and k eK] is a subgroup of G. What if G is abelian? 48. Let G be a group and g E G. Show that E(G)-(x E G : gx = xg for all g E G} .08 is a subgroup of G. This subgroup is called the center of G. 49. Let a and b be elements of a group G. If atb ba and a3 -e, prove that ab ba 50. Give an example of an infinite group in which every nontrivial subgroup is infinite. 51. If zy = x-ly-1 for all x and y in G, prove that G must be abelian. 52. Prove or disprove: Every proper subgroup of an nonabelian group is nonabelian. 53. Let H be a subgroup of G and [, 1 C(H)={g e G : gh = hg for all hem. Prove C(H) is a subgroup of G. This subgroup is called the centralizer of H in G. 54. Let HH be a subgroup of G. If g E G, sho hat gHHgghg subgroup of G. :h e Hy is also a 3.5 Additional Exercises: Detecting Errors 1. (UPC Symbols) Universal Product Code (UPc) symbols are found on most products in grocery and retail stores. The UPC symbol is a 12-digit code identifying the manufacturer of a product and the product itself (Figure 3.32). The first 1l digits contain information about the product; the twelfth digit is used for error detection. If dd.di2 is a valid UPC number, then 3 di +1 d2+3d+3 di+1 di2 (mod 10) 332is a valid

Explanation / Answer

xy=x-1.y-1 implies that (xy) ^2=e ,that in this every element has order 2. This implies, xy. xy=e,( where e is identity of a group G.). This implies xy. xy. yx=e. yx(multiply both side by yx).

This implies, xy. x. (y. y). x=yx

This implies xy. x. e. x=yx ( since, y. y=y^2=e).

This implies xy. (x. x) =yx

This implies xy. e=yx (since x. x=x^2=e)

This implies xy=yx for all x €G. This shows that G is abelian.

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