Cutting subgroup

Definition with symbols
A subgroup $$H$$ of $$G$$ is said to be a cutting subgroup if it is a self-normalizing subgroup and satisfies the further conditions (equivalent subject to the subgroup being self-normalizing):


 * There exists a real representation $$V$$ of $$G$$ such that the dimension of $$V^H$$ is one more than the dimension of $$V^G$$ ($$V^H$$ and $$V^G$$ are the subspaces comprising points fixed pointwise under the action of $$H$$ and $$G$$ respectively)
 * There exists a real representation $$V$$ of $$G$$ such that $$V^G$$ has dimension 0 and $$V^H$$ has dimension 1
 * There exists an irreducible nontrivial real representation $$V$$ of $$G$$ such that $$V^H$$ has dimension 1