# Residually operator

*This article defines a group property modifier (a unary group property operator) -- viz an operator that takes as input a group property and outputs a group property*

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*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:*residual group property

## Definition

### Symbol-free definition

The residually operator is a map from the group property space to itself that takes as input a group property and outputs the property of being a group such that:

For any nonidentity element of the group, there exists a normal subgroup *not* containing that element such that the quotient group has property .

### Definition with symbols

Let be a group property. A group is said to be **residually** if it satisfies the following equivalent conditions:

- For any non-identity element , there exists a normal subgroup of such that and satisfies property .
- is a subdirect product of a collection of groups, all of which satisfy property .

## Application

Important instances of application of the residually operator:

- residually nilpotent group: obtained from nilpotent group
- residually finite group: obtained from finite group