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
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 .
Important instances of application of the residually operator: