Core-free operator

This article defines a subgroup property modifier (a unary subgroup property operator) -- viz an operator that takes as input a subgroup property and outputs a subgroup property

View a complete list of subgroup property modifiers OR View a list of all subgroup property operators (possibly with multiple inputs)

Definition

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