Cartan subalgebras and related notion in Group Theory According to Humphreys, a Cartan sub-algebra of a Lie algebra $L$ is a nilpotent Lie sub-algebra whose normalizer is itself.
Look at analogous things in groups, or even in just finite groups.
If $G$ is a finite group, and if $H$ is a subgroup such that $H$ is nilpotent and is equal to its normalizer in $G$, what such subgroups are called? Have such subgroups studied? Is their existence in all finite groups studied/known?

Nilpotent self-normalising subgroups are called Carter-subgroups. Carter (1961) proved that any finite solvable group has a Carter subgroup, and all its Carter subgroups are conjugate subgroups (and therefore isomorphic). If a group is not solvable it need not have any Carter subgroups: for example, the alternating group $A_5$ of order $60$ has no Carter subgroups.   
For linear groups one could consider Borel subgroups, which are self-normalizing.
They are solvable but not nilpotent. But then one could consider a maximal torus, which is "almost" self-normalizing.
