Abstract Algebra Theory and Applications - Computer Science ...

50 CHAPTER 2 GROUPS3. Write out Cayley tables for groups formed by the symmetries of a rectangleand for (Z 4 , +). How many elements are in each group? Are the groups thesame? Why or why not?4. Describe the symmetries of a rhombus and prove that the set of symmetriesforms a group. Give Cayley tables for both the symmetries of a rectangleand the symmetries of a rhombus. Are the symmetries of a rectangle andthose of a rhombus the same?5. Describe the symmetries of a square and prove that the set of symmetriesis a group. Give a Cayley table for the symmetries. How many ways canthe vertices of a square be permuted? Is each permutation necessarily asymmetry of the square? The symmetry group of the square is denoted byD 4 .6. Give a multiplication table for the group U(12).7. Let S = R \ {−1} and define a binary operation on S by a ∗ b = a + b + ab.Prove that (S, ∗) is an abelian group.8. Give an example of two elements A and B in GL 2 (R) with AB ≠ BA.9. Prove that the product of two matrices in SL 2 (R) has determinant one.10. Prove that the set of matrices of the form⎛1 x y⎞⎝ 0 1 z ⎠0 0 1is a group under matrix multiplication. This group, known as the Heisenberggroup, is important in quantum physics. Matrix multiplication in theHeisenberg group is defined by⎛⎝1 x y0 1 z0 0 1⎞ ⎛⎠ ⎝1 x ′ y ′0 1 z ′0 0 1⎞⎛⎠ = ⎝1 x + x ′ y + y ′ + xz ′0 1 z + z ′0 0 111. Prove that det(AB) = det(A) det(B) in GL 2 (R). Use this result to showthat the binary operation in the group GL 2 (R) is closed; that is, if A and Bare in GL 2 (R), then AB ∈ GL 2 (R).12. Let Z n 2 = {(a 1 , a 2 , . . . , a n ) : a i ∈ Z 2 }. Define a binary operation on Z n 2 by(a 1 , a 2 , . . . , a n ) + (b 1 , b 2 , . . . , b n ) = (a 1 + b 1 , a 2 + b 2 , . . . , a n + b n ).Prove that Z n 2 is a group under this operation. This group is important inalgebraic coding theory.13. Show that R ∗ = R \ {0} is a group under the operation of multiplication.⎞⎠ .