| Let $K$ be a compact group, and let $t\in K$ be an element whose centralizer has minimal dimension (such elements are dense in $K$). Let $T$ be the centralizer of $t$. This subgroup is closed since $T=\vp^{-1}(t)$ where $\vp:K\to K$ is the map $k\mapsto ktk^{-1}$, and abelian since it is the intersection of $K$ with the Cartan subgroup of its complexification, and hence a torus, since $K$ (and thus $T$) is compact. We call $T$ a {\em maximal torus} of $K$. |
Let $K$ be a compact group, and let $t\in K$ be an element whose centralizer has minimal dimension (such elements are dense in $K$). Let $T$ be the centralizer of $t$. This subgroup is closed since $T=\vp^{-1}(t)$ where $\vp:K\to K$ is the map $k\mapsto ktk^{-1}$, and abelian since it is the intersection of $K$ with the Cartan subgroup of its complexification, and hence a torus, since $K$ (and thus $T$) is compact. We call $T$ a maximal torus of $K$. |
