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

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
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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 $p$ 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 $p$ .

Definition with symbols

Let $\alpha$ be a group property. A group $G$ is said to be residually $\alpha$ if it satisfies the following equivalent conditions:

For any non-identity element $g\in G$ , there exists a normal subgroup $N$ of $G$ such that $g\notin N$ and $G/N$ satisfies property $\alpha$ .
$G$ is a subdirect product of a collection of groups, all of which satisfy property $\alpha$ .