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 $K$ .

This term is also applied to the corresponding maximal abelian subgroup of a complex semisimple group, which is an algebraic torus.