Core-free operator

Symbol-free definition
Let $$p$$ be a strongly join-closed subgroup property, viz $$p$$ is always true for the trivial subgroup and further, an arbitrary join of subgroups with proeprty $$p$$ also has property $$p$$.

Then, the core-free operator applied to $$p$$ gives the following property: a subgroup satisfies this property in the group if there is no nontrivial subgroup contained inside that that satisfies $$p$$. Equivalently, a subgroup is $$p$$-core-free if the $$p$$-core of this subgroup is the trivial subgroup.

Definition with symbols
Let $$p$$ be a strongly join-closed subgroup property, viz $$p$$ is always true for the trivial subgroup and further, an arbitrary join of subgroups with proeprty $$p$$ also has property $$p$$.

Then, the core-free operator applied to $$p$$ gives the following property $$q$$. $$H$$ satisfies $$q$$ in $$G$$ if for any nontrivial subgroup $$K$$ of $$H$$, $$K$$ does not satisfy $$p$$ in $$G$$.

Facts
If $$p$$ is also a trim subgroup property then any proper subgroup of a $$p$$-simple group must be $$p$$-core-free. Thu