Simple group operator

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Template:Subgroup-to-group property operator

Definition

Symbol-free definition

The simple group operator is an operator from the collection of trim subgroup properties to the group property space. It acts as follows: given a trim subgroup property p, a group is termed p-simple if it is nontrivial and has no proper nontrivial subgroup satisfying p.

(A subgroup property is trim if it is satisfied by both the whole group and the trivial subgroup).

Note that the trivial group is not considered p-simple.

Examples

Facts

Simple-complete property

A trim property p is termed simple-complete if every group can be embedded as a proper subgroup of a p-simple group.

Core operator

Suppose the given property p is trim and join-closed. Then, in a p-simple group, the p-core of any proper subgroup is trivial. That is, every proper subgroup is p-core-free.

If p is a simple-complete property, then every subgroup of a group is potentially core-free with respect to p.

Closure operator

Suppose p is intersection-closed and trim. Then, in a p-simple group, the p-closure of any nontrivial subgroup is the whole group. Equivalently, every nontrivial group is contra-p.

Thus, if p is a simple-complete property, then every nontrivial subgroup of a group is 'potentially contrap.