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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)
This property modifier is idempotent and a property is a fixed-point, or equivalently, an image of this if and only if it is a:intermediate subgroup condition
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The potentially operator is a map from the subgroup property space to itself defined as follows. Given a subgroup property , the subgroup property potentially is defined as the property of being a subgroup in a group such that there exists some group containingthe bigger group, in which the subgroup has property .
Definition with symbols
has property potentially in if there exists a group containing such that has property in .
If every subgroup satisfying property also satisfies , then every subgroup satisfying potentially must also satisfy potentially .
In particular, if satisfies the intermediate subgroup condition (and is hence invariant under the potentially operator), then implies that potentially .
Also, if is potential-tautological, then must also be potential-tautological.
Any subgroup which satisfies property must also satisfy potentially .
The potentially operator is idempotent, and the subgroup properties that are invariant under this operator are precisely the subgroup properties that satisfy the intermediate subgroup condition.
Effect on metaproperties
Properties obtained via this operator
The most important of properties obtained dircetly via this operator is the property of being a potentially characteristic subgroup.