Note on Quaternions

ON QUATERNIONS By

Understanding quaternions

The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared for its importance with the invention of triple coordinates by Descartes. The ideas of this calculus, as distinguished from its operations and symbols, are fitted to be of the greatest use in all parts of science.-Clerk Maxwell, 1869. Quaternions came from Hamil...

Generalized Quaternions

The quaternion group Q8 is one of the two non-abelian groups of size 8 (up to isomorphism). The other one, D4, can be constructed as a semi-direct product: D4 ∼= Aff(Z/(4)) ∼= Z/(4) o (Z/(4))× ∼= Z/(4) o Z/(2), where the elements of Z/(2) act on Z/(4) as the identity and negation. While Q8 is not a semi-direct product, it can be constructed as the quotient group of a semi-direct product. We wil...

On Infinite Groups Generated by Two Quaternions

Let x, y be two integer quaternions of norm p and l, respectively, where p, l are distinct odd prime numbers. What can be said about the structure of 〈x, y〉, the multiplicative group generated by x and y ? Under a certain condition which excludes 〈x, y〉 from being free or abelian, we show for example that 〈x, y〉, its center, commutator subgroup and abelianization are finitely presented infinite...

Some Notes on Unit Quaternions and Rotation