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
- 1 Definition
- 2 Properties
- 3 Effect on metaproperties
- 4 Properties obtained via this operator
The intermediately operator is a map from the subgroup property space to itself, that sends a subgroup property to the property of being a subgroup that satisfies not only in the whole group, but also in every intermediate subgroup.
Definition with symbols
Given a subgroup property , the subgroup property intermediately is the property as follows: satisfies intermediately in if for any group with , satisfies in .
If are two subgroup properties, then intermediately intermediately . This follows directly from the definition.
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 , intermediately . This follows from the fact that if satisfies property in every intermediate subgroup, also satisfies property in the whole group.
The intermediately operator is idempotent, in the sense that applying the intermedaitely operator twice has the same effect as applying it once. The image-cum-fixed-point-space for this operator is precisely the subgroup properties satisfying the intermediate subgroup condition.
Effect on metaproperties
Suppose is a join-closed subgroup property, viz the join of any family of subgroups satisfying property , also satisfies property . Then, it is easy to see that intermediately is also join-closed.
It is not clear whether, even if is transitive, intermediately will b transitive.
If satisfies the transfer condition, it also, in particular, satisfies the intermediate subgroup condition, and hence is unchanged under application of the intermediately operator.
Properties obtained via this operator
Naturally arising properties that satisfy intermediate subgroup condition