Hereditarily operator

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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.

Examples

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