# Subnormal depth

From Groupprops

## Definition

The **subnormal depth** (also sometimes called the **defect** or **subnormal defect**) of a subnormal subgroup in a group is defined in the following equivalent ways:

- It is the smallest for which there exists an ascending chain of subgroups where each is normal in .
- Consider the sequence , is the normal closure of in . The subnormal depth is the smallest for which .
- Consider the sequence where and . The subnormal depth is the smallest for which .

We typically say that a subgroup has subnormal depth if its subnormal depth is less than or equal to . A subgroup of subnormal depth (less than or equal to) is termed a -subnormal subgroup.

### Equivalence of definitions

`Further information: Equivalence of definitions of subnormal subgroup`

## Particular cases

- A subgroup has subnormal depth if and only if it is the whole group.
- A subgroup has subnormal depth (less than or equal to) if and only if it is a normal subgroup.
- A subgroup has subnormal depth (less than or equal to) if and only if it is a 2-subnormal subgroup.
- A subgroup has subnormal depth (less than or equal to) if and only if it is a 3-subnormal subgroup.
- A subgroup has subnormal depth (less than or equal to) if and only if it is a 4-subnormal subgroup.