Minimal 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 propertyView a complete list of subgroup property modifiers OR View a list of all subgroup property operators (possibly with multiple inputs)
Definition
The minimal operator is a subgroup property modifier that takes as input a subgroup property and outputs a subgroup property
defined as follows:
has property
in
if the following three hold:
-
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
.
Application
Some important instances of application of the maximal operator:
- minimal normal subgroup: obtained from normal subgroup
- minimal characteristic subgroup: obtained from characteristic subgroup
Properties
Monotonicity
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
.
Descendance
This subgroup property modifier is descendant, viz the image of any subgroup property under this modifier is stronger than that property. In symbols, if denotes the modifier and
and property,
For any subgroup property , the property of being a minimal
-subgroup, is stronger than the property of being a
-subgroup.
Idempotence
This subgroup property modifier is idempotent, viz applying it twice to a subgroup property has the same effect as applying it once
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.