Contra 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 intersection-closed subgroup property. In other words, an arbitrary intersection of subgroups with property $p$ also has property $p$, and further, any group has property $p$ as a subgroup of itself.

Then the contra-property to $p$, viz the property obtained by applying the contra operator to $p$ is defined as the property of being a subgroup such that there is no proper subgroup containing it that satisfies $p$.

Definition with symbols

Let $p$ be a strongly intersection-closed subgroup property. In other words, an arbitrary intersection of subgroups with property $p$ also has property $p$, and further, any group has property $p$ as a subgroup of itself.

Then the contra-property to $p$, viz the property obtained by applying the contra operator to $p$ is defined as the following property $q$: a subgroup $H$ satisfies $q$ in $G$ if there is no proper subgroup $K$ of $G$ containing $H$, for which $K$ satisfies $p$ in $G$.

Facts

If $p$ is a trim subgroup property then any $p$-simple group has the property that every nontrivial subgroup satisfies contra-$p$.

In particular, if $p$ is a simple-complete subgroup property, then every nontrivial subgroup is potentially contra-$p$.