1.1.4 Special Linear Group

The gen­er­al linear group $GL(n,F)$ con­tains el­e­ments with non-zero de­ter­mi­nants, and there exists a sub­group of $GL(n,F)$ with el­e­ments having de­ter­mi­nants equal to one. This sub­group has a name, the spe­cial linear group, where the term spe­cial de­notes de­ter­mi­nants equal to $1$. The spe­cial linear group is the sub­group of $GL(n,F)$ with in­vert­ible linear $n \times n$ ma­tri­ces over the field F, with degree $n$, and de­ter­mi­nants equal to $1$. The group and its defin­ing prop­er­ties can then be ex­pressed using the fol­low­ing no­ta­tion