## Franklin's Notes

### Matrix groups and linear groups

Given a field $\mathbb k$, the set of full-rank $n\times n$ matrices over $\mathbb k$ constitute a group whose group operation is matrix multiplication. This group is called the general linear group of dimension $n$ over $\mathbb k$, and is denoted $\mathrm{GL}_n(\mathbb k)$. In general, a matrix group is a subgroup of $\mathrm{GL}_n(\mathbb k)$, and a linear group is any group $G$ that is isomorphic to a matrix group. Here's a list of common examples of matrix groups:

• The general linear group $\mathrm{GL}_n(\mathbb k)$

• The special linear group $\mathrm{SL}_n(\mathbb k)$, consisting of the $n\times n$ matrices over $\mathbb k$ with determinant equal to $1$

• The projective special linear group $\mathrm{PSL}_n(\mathbb k)$ is the quotient group of $\mathrm{SL}_n(\mathbb k)$ by its normal subgroup of scalar transformations

• The special orthogonal group $\mathrm{SO}_n(\mathbb k)$ consists of the group of $n\times n$ orthogonal matrices with determinant $1$

• The group of invertible $n\times n$ upper-triangular (or lower-triangular) matrices over $\mathbb k$

Given a polynomial or a form $f\in \mathbb k[x]$ in $n$ variables, we may also consider the subgroup of $A\in \mathrm{GL}_n(\mathbb k)$ satisfying $f(\vec x)=f(A\vec x)$. This is the stabilizer with respect to $f$ of the pre-composition group action of $\mathrm{GL}_n(\mathbb k)$ on $\mathbb k[x]$.