Normal core of normalizer

This article defines a subgroup operator related to the subgroup property normal subgroup. By subgroup operator is meant an operator that takes as input a subgroup of a group and outputs a subgroup of the same group.

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 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 $G$. Further information: 2-subnormal subgroup has a unique fastest ascending subnormal series