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)
- has property in
- is a nontrivial subgroup of
- There is no nontrivial subgroup of , contained in , such that satisfies .
In other words, is a minimal element (in the containment partial order) among those nontrivial subgroups of that satisfy .
Some important instances of application of the maximal operator:
- minimal normal subgroup: obtained from normal subgroup
- minimal characteristic subgroup: obtained from characteristic subgroup
The minimal operator is not monotone. In other words, if are subgroup properties, then a maximal -subgroup need not be a maximal -subgroup. The reason is that there may be smaller subgroups that satisfy property but not property .
For any subgroup property , the property of being a minimal -subgroup, is stronger than the property of being a -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.