# 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 property

View 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 $p$, the property hereditarily $p$ is defined as the property of being a subgroup of a group such that every subgroup contained in it satisfies property $p$ in the whole group.

### Definition with symbols

Let $p$ be a subgroup property. The property hereditarily $p$ is defined as follows. A subgroup $H$ satisfies hereditarily $p$ in a group $G$ if for every subgroup $K \le H$, $K$ satisfies property $p$ in $G$.

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