Notes - Groups HT23, Special groups

Flashcards

@Define the “general linear” group $\mathrm{GL} _ n(\mathbb R)$.

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

@Define the “special linear” group $\mathrm{SL} _ n(\mathbb R)$.

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

@Define the “orthogonal” group $\mathrm O _ n(\mathbb R)$.

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

@Define the “special orthogonal” group $\mathrm{SO} _ n(\mathbb R)$.

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

@Define 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)?

\[2n\]

@Define 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

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

Todo.

@todo~