# Normal core of normalizer

From Groupprops

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 of a group is defined in the following equivalent ways:

- It is the largest normal subgroup of that normalizes . In other words, it is the largest normal subgroup of contained in the normalizer .
- It is the normal core of the normalizer in .

is contained in the normal core of normalizer of if and only if is a 2-subnormal subgroup of . In this case, if is the normal core of normalizer of , then the ascending chain is the unique fastest ascending subnormal series for in . `Further information: 2-subnormal subgroup has a unique fastest ascending subnormal series`