Residually operator

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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:

  1. 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.
  2. G is a subdirect product of a collection of groups, all of which satisfy property \alpha.

Application

Important instances of application of the residually operator: