Difference between revisions of "Upwardclosure operator"
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Revision as of 00:34, 8 May 2008
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
Symbolfree 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 upwardclosed subgroup property.
Template:Monotone subgroup property modifier
If (both are subgroup properties) then the where denotes the upward closure.