# Hereditarily operator

This article defines a subgroup property modifier (a unary subgroup property operator) -- viz an operator that takes as input a subgroup property and outputs a subgroup propertyView a complete list of subgroup property modifiers OR View a list of all subgroup property operators (possibly with multiple inputs)

*This property modifier is idempotent and a property is a fixed-point, or equivalently, an image of this if and only if it is a:*left-hereditary subgroup property

## Definition

### Symbol-free definition

The **hereditarily operator**, also called the **left-hereditarily operator**, is defined as follows: given a subgroup property , the property hereditarily is defined as the property of being a subgroup of a group such that every subgroup contained in it satisfies property in the whole group.

### Definition with symbols

Let be a subgroup property. The property **hereditarily** is defined as follows. A subgroup satisfies hereditarily in a group if for every subgroup , satisfies property in .

A subgroup property obtained by applying this operator is a left-hereditary subgroup property.

## Examples

- Hereditarily normal subgroup: A subgroup such that every subgroup of it is normal in the whole group. Hereditarily normal subgroups are also termed
*quasicentral*. Note that any central subgroup (i.e., any subgroup contained in the center) is hereditarily normal. - Hereditarily permutable subgroup: A subgroup such that every subgroup of it is permutable in the whole group. The Baer norm of a group is a hereditarily permutable subgroup.
`For full proof, refer: Baer norm is hereditarily permutable`.

For more examples, see Category:Properties obtained by applying the hereditarily operator or Category:Left-hereditary subgroup properties.