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 $K\leq H$, $K$ satisfies property $p$ in $G$.

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.