Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 14910. Show that the matrices⎛1 0 0⎞ ⎛⎝ 0 1 0 ⎠ ⎝0 0 1form a group.order 6.⎛⎝0 0 11 0 00 1 0⎞ ⎛⎠ ⎝1 0 00 0 10 1 00 0 10 1 01 0 0⎞ ⎛⎠ ⎝⎞ ⎛⎠ ⎝0 1 01 0 00 0 10 1 00 0 11 0 0Find an isomorphism of G with a more familiar group of11. Find five non-isomorphic groups of order 8.12. Prove S 4 is not isomorphic to D 12 .13. Let ω = cis (2πi/n) be a primitive nth root of unity. Prove that the matrices( ) ω 0A =0 ω −1andB =( 0 11 0form a multiplicative group isomorphic to D n .14. Show that the set of all matrices of the form( )±1 nB =,0 1where n ∈ Z n , is a group isomorphic to D n .15. List all of the elements of Z 4 × Z 2 .16. Find the order of each of the following elements.(a) (3, 4) in Z 4 × Z 6(b) (6, 15, 4) in Z 30 × Z 45 × Z 24(c) (5, 10, 15) in Z 25 × Z 25 × Z 25(d) (8, 8, 8) in Z 10 × Z 24 × Z 8017. Prove that D 4 cannot be the internal direct product of two of its propersubgroups.18. Prove that the subgroup of Q ∗ consisting of elements of the form 2 m 3 n form, n ∈ Z is an internal direct product isomorphic to Z × Z.19. Prove that S 3 × Z 2 is isomorphic to D 6 . Can you make a conjecture aboutD 2n ? Prove your conjecture. [Hint: Draw the picture.])⎞⎠⎞⎠