Intermediately 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)
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
Contents
Definition
Symbol-free definition
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
.
Properties
Monotonicity
This subgroup property modifier is monotone, viz if are subgroup properties and
is the operator, then
If are two subgroup properties, then intermediately
intermediately
. This follows directly from the definition.
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
, intermediately
. This follows from the fact that if
satisfies property
in every intermediate subgroup,
also satisfies property
in the whole group.
Idempotence
This subgroup property modifier is idempotent, viz applying it twice to a subgroup property has the same effect as applying it once
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
Template:Join-closedness-preserving
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.
Transitivity
It is not clear whether, even if is transitive, intermediately
will b transitive.
Transfer condition
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
These include: