Notes - Groups HT23, Special groups

What is the “general linear” group $\text{GL} _ n(\mathbb R)$?

The group of all invertible $n \times n$ matrices.

What is the “special linear” group $\text{SL} _ n(\mathbb R)$?

The group of all invertible $n \times n$ matrices with determinant $1$.

What is the “orthogonal” group $\text O (\mathbb R)$?

The group of all orthogonal $n \times n$ matrices.

What is the “special orthogonal” group $\text {SO} _ n(\mathbb R)$?

The group of all orthogonal $n \times n$ matrices with determinant $1$.

What is the dihedral group $D _ {2n}$?

The group of isometries under composition of a regular $n$-gon in the plane.

Can you list all the elements of the dihedral group $D _ 8$?

\[\\{e, \rho, \rho^2, \rho^3, s, \rho s, \rho^2 s, \rho^3 s\\}\]

What is the order of the group containing all the isometries of a regular $n$-gon in the plane (the dihedral group)?

\[2n\]

What is the group $Q _ 8$?

The quaternion group

\[\\{\pm 1, \pm \pmb i, \pm \pmb j, \pm \pmb k \\}\]

Prove that $D _ {2n}$ has $2n$ elements, namely

\[\\{e, \rho, \cdots, \rho^n, s, \rho s, \cdots, \rho^{n-1} s\\}\]

Todo.