Determine with proof whether the following statements are tr

Determine with proof whether the following statements are true or false. (a) There exists a non-abelian group of infinite order.

Solution

Quaternions are generally represented in the form:

a + bi + cj + dk

where a, b, c, and d are real numbers, and i, j, and k are the fundamental quaternion units.

Quaternion multiplication

we notice that i . j= k and j.i=-k the set of all quarternions will not satisfy commutative property and hence not abelian   

the set is of infinite order

multiplication

×	1	i	j	k
1	1	i	j	k
i	i	1	k	j
j	j	k	1	i
k	k	j	i	1