completed program expository paper · long island research program

Algebraic Properties of Quaternions

I wrote this expository paper to understand what changes when multiplication stops commuting. It starts with the eight-element quaternion group and then moves to the full quaternion algebra.

Scope: The group structure and inverse formula are established mathematics. I am not presenting this as a new theorem, an original research result, or a publication.

the question I worked through

The first half studies a set with eight elements e, θ, a, b, c, θa, θb, θc. Under the correspondence e ↦ 1, θ ↦ −1, a ↦ i, b ↦ j, c ↦ k, it is the quaternion group Q8. I calculated its products and asked which elements commute with everything.

The second half moves from that finite group to Hamilton's quaternions H. I worked through addition, noncommutative multiplication, conjugation, the norm, and the inverse of a nonzero quaternion.

Key relations: a2 = b2 = c2 = θ, ab = c, and ba = θc. The last two products show immediately that order matters.

the main arguments

  • calculating the Cayley table

    I used the defining relations to calculate every product in Q8. The table makes closure, identities, inverses, and noncommutativity visible in one place.

  • finding the center

    The identity and θ commute with all eight elements. Each of the other six fails to commute with at least one generator, so in standard notation Z(Q8) = {±1}.

  • showing multiplication is noncommutative

    In H, ij = k while ji = −k. Quaternion addition is commutative, but quaternion multiplication is not, so H is a noncommutative ring.

  • working out the inverse

    For q = a + bi + cj + dk, I used the conjugate q̄ = a − bi − cj − dk and the positive norm N(q) = a2 + b2 + c2 + d2. When q ≠ 0, the standard formula q−1 = q̄ / N(q) gives a two-sided inverse. This is why H is a division ring.

what I would strengthen

The weakest part of my submitted argument is associativity. I checked examples and then relied on the identification with Q8 without fully establishing it. In a revision, I would define the eight elements directly inside H, where associativity is already known, and then verify that the displayed relations match.

I would also separate two classification results that the draft blurred together: the real finite-dimensional associative division algebras are R, C, and H, while the octonions are nonassociative and are not a ring in the usual sense.

The application I would keep is narrower: unit quaternions are widely used to represent and interpolate 3D orientations in graphics, robotics, and aerospace. That continuous set is different from the finite group Q8 studied at the start.

sources from the paper