Upward-closure 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)
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Definition
Symbol-free definition
Let be a subgroup property. Then the upward closure of
is defined as the property of being a subgroup such that all subgroups containing it satisfy property
in the whole group.
Definition with symbols
Let be a subgroup property. Then, the upward closure of
is defined as the following subgroup property
: A subgroup
satisfies property
in
, if for every subgroup
with
,
satisfies
in
.
Properties
Template:Idempotent subgroup property modifier
Applying the upward closure operator twice is the same as applying it once. In other words, the properties that are fixed under the upward closure operator are precisely the same as the properties that can be obtained as images of the upward closure operator. A property that is fixed under the upward closure operator is termed an upward-closed subgroup property.
Template:Monotone subgroup property modifier
If (both are subgroup properties) then the
where
denotes the upward closure.