Residually operator

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: