Normal core of normalizer

Definition
The normal core of normalizer of a subgroup $$H$$ of a group $$G$$ is defined in the following equivalent ways:


 * 1) It is the largest normal subgroup of $$G$$ that normalizes $$H$$. In other words, it is the largest normal subgroup of $$G$$ contained in the normalizer $$N_G(H)$$.
 * 2) It is the defining ingredient::normal core of the normalizer $$N_G(H)$$ in $$G$$.

$$H$$ is contained in the normal core of normalizer of $$H$$ if and only if $$H$$ is a 2-subnormal subgroup of $$G$$. In this case, if $$K$$ is the normal core of normalizer of $$H$$, then the ascending chain $$H \le K \le G$$ is the unique fastest ascending subnormal series for $$H$$ in $$G$$.