Quasi-normal subgroups of algebraic groups (Q1072649)

Let G be a connected linear algebraic group over an algebraically closed field. A subgroup H of G is said to be algebraically quasi-normal in G if it is Zariski-closed and the closure of AH and HA coincide, for any closed subgroup A of G. Clearly every closed normal subgroup H of G is algebraically quasi-normal. The author proves the converse for fields of characteristic zero and he also offers a complete discussion in the case of positive characteristic.