If you find any mistakes, please make a comment! Thank you.

## The quaternion group is not a subgroup of Symmetric group for any n less than 8

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 4.2 Exercise 4.2.7 Solution: (1) $Q_8$ is a subgroup of $S_8$ via the left regular representation. (2) Now suppose…

## Embed the Hamiltonian quaternions in a ring of real matrices

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.14 Solution: Define $\varphi : \mathbb{H} \rightarrow M_4(\mathbb{R})$ as follows. a+bi+cj+dk \mapsto \begin{bmatrix} a & b &…

## Embed a ring of quadratic integers in a ring of matrices

Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.12 Solution: We begin with a lemma. Lemma: If $D \in \mathbb{Z}$ is not a perfect square,…