Let G = {±1, ±i, ±j, ±k}, where i 2 = j2 = k2 = –1, –i = (–1)i, 12 = (–1)2 = 1, ij = –ji...

Let G = {±1, ±i, ±j, ±k}, where i ² = j² = k² = –1, –i = (–1)i, 1² = (–1)² = 1, ij = –ji = k, jk = –kj = i, and ki = –ik = j.

a. Construct the Cayley table for G.

b. Show that .

c. Construct the Cayley table for G/H. Is G/H isomorphic to Z4 or (The rules involving i, j, and k can be remembered by using the circle below.

Going clockwise, the product of two consecutive elements is the third one. The same is true for going counterclockwise, except that we obtain the negative of the third element.) This is the group of quaternions that was given in another form in Exercise 4 in the Supplementary Exercises for Chapters 1–4. It was invented by William Hamilton in 1843. The quaternions are used to describe rotations in three dimensional space, and they are used in physics. The quaternions can be used to extend the complex numbers in a natural way.