Linear algebraic group

Definition

Definition with symbols

A linear algebraic group over a field $k$ is defined as an abstract group $G$, along with an embedding of $G$ as a closed subgroup of $GL(n,k)$ for some choice of $n$. Note that such an embedding automatically turns $G$ into an algebraic group, because any closed subgroup of an algebraic group automatically inherits the structure of an algebraic group.