Minimal operator

Definition
The minimal operator is a subgroup property modifier that takes as input a subgroup property $$p$$ and outputs a subgroup property $$q$$ defined as follows: $$H$$ has property $$q$$ in $$G$$ if the following three hold:


 * $$H$$ has property $$p$$ in $$G$$
 * $$H$$is a nontrivial subgroup of $$G$$
 * There is no nontrivial subgroup $$K$$ of $$G$$, contained in $$H$$, such that $$K$$ satisfies $$p$$.

In other words, $$H$$ is a minimal element (in the containment partial order) among those nontrivial subgroups of $$G$$ that satisfy $$p$$.

Application
Some important instances of application of the maximal operator:



Monotonicity
The minimal operator is not monotone. In other words, if $$p \le q$$ are subgroup properties, then a maximal $$p$$-subgroup need not be a maximal $$q$$-subgroup. The reason is that there may be smaller subgroups that satisfy property $$q$$ but not property $$p$$.

For any subgroup property $$p$$, the property of being a minimal $$p$$-subgroup, is stronger than the property of being a $$p$$-subgroup.

Clearly, applying the maximal operator twice has the same effect as applying it once. Those subgroup properties that are obtained by applying this operator are termed minimal subgroup properties.