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Problems in Group Theory and Quaternions

1. Define the ring of quaternions H := { a1 + bi + cj + dk : a, b, c, d <- R }, with the relations
1 = 1 and i^2 = j^2 = k^2 = ijk = -1. Define the quaternion absolute value by
|a1 + bi + cj + dk|^2 := a^2 + b^2 + c^2 + d^2 .

Note H is actually isomorphic to R^4 as a vector space, but it has more structure than R^4.

(a) Given q = a1 + bi + cj + dk, define the quaternion conjugate qbar := a1 - bi - cj - dk and show that (q qbar) = |q|^2 .
(b) Show that H is actually a division algebra by finding the inverse of q = a1 + bi + cj + dk . Note that H is not a field because it is not commutative.
(c) The nonzero quaternions H^x are isomorphic to a subgroup of GL2(C) via the map

a1 + bi + cj + dk <-> ( a + id -b -ic)
( b - ic a - id) .

Use this to prove that |uv| = |u||v| for all u, v <- H

[The quaternions were discovered by William Rowan Hamilton on October 16, 1843, as he was walking with his wife along the Royal Canal in Dublin. To celebrate the discovery, he immediately carved this equation into the stone of the Brougham Bridge: i^2 = j^2 = k^2 = ijk = -1.]

2. Recall that Z/nZ has a unique (cyclic) subgroup of order d for each d|n. Please refer to the attachment for complete question.

4. Explicitly describe the conjugacy classes of the Dihedral group

Dn := <r, p : r^2 = p^n, pr = rp^-1>.

Hint: Every element of Dn looks like rp^k or p^k for some k.


Solution Preview

Please see the attachment for solution, as it contains lot of equation (images).

1. (a) Given the quaternion and its conjugate , show that .
First we note that the defining relation for the unit quaternions, namely

implies the following multiplication table:


Solution Summary

We solve three problems involving quaternions as well as dihedral and cyclic groups.