Normal subgroup of characteristic subgroup

Symbol-free definition
A subgroup of a group is termed a normal subgroup of characteristic subgroup if it satisfies the following equivalent conditions:


 * 1) It is a defining ingredient::normal subgroup of a defining ingredient::characteristic subgroup of the group.
 * 2) It is normal inside its defining ingredient::characteristic closure in the group.
 * 3) Its defining ingredient::characteristic closure is contained in its normalizer.
 * 4) It is contained in the defining ingredient::characteristic core of its normalizer.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed a normal subgroup of characteristic subgroup if it satisfies the following equivalent conditions:


 * 1) There exists a characteristic subgroup $$K$$ of $$G$$ such that $$H$$ is a normal subgroup of $$K$$.
 * 2) $$H$$ is a normal subgroup inside the characteristic closure of $$H$$ in $$G$$.
 * 3) The characteristic closure of $$H$$ in $$G$$ is contained in the normalizer $$N_G(H)$$.
 * 4) $$H$$ is contained in the characteristic core of $$N_G(H)$$ in $$G$$.

Facts

 * Left residual of 2-subnormal by normal is normal of characteristic: If $$H$$ is a subgroup of $$K$$ with the property that whenever $$K$$ is normal in a group $$G$$, $$H$$ is 2-subnormal in $$G$$, then $$H$$ is a normal subgroup of characteristic subgroup in $$K$$.