Simple group 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

• Normal subgroup gives rise to simple group
• Characteristic subgroup gives rise to characteristically simple group
• Direct factor gives rise to directly indecomposable group
• Retract gives rise to semidirectly indecomposable group
• Ascendant subgroup gives rise to strictly simple group
• Serial subgroup gives rise to absolutely simple group

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 contra$p$.