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 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

Definition

Symbol-free definition

The intermediately operator is a map from the subgroup property space to itself, that sends a subgroup property $p$ to the property of being a subgroup that satisfies $p$ not only in the whole group, but also in every intermediate subgroup.

Definition with symbols

Given a subgroup property $p$, the subgroup property intermediately $p$ is the property as follows: $H$ satisfies intermediately $p$ in $G$ if for any group $K$ with $H\le K\le G$, $H$ satisfies $p$ in $K$.

Operator properties

If $p\le q$ are two subgroup properties, then intermediately $p\le$ intermediately $q$. This follows directly from the definition.

Template:Descendant operator For any subgroup property $p$, intermediately $p\le p$. This follows from the fact that if $H$ satisfies property $p$ in every intermediate subgroup, $H$ also satisfies property $p$ 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 $p$ is a join-closed subgroup property, viz the join of any family of subgroups satisfying property $p$, also satisfies property $p$. Then, it is easy to see that intermediately $p$ is also join-closed.

Transitivity

It is not clear whether, even if $p$ is transitive, intermediately $p$ will b transitive.

Transfer condition

If $p$ satisfies the transfer condition, it also, in particular, satisfies the intermediate subgroup condition, and hence $p$ is unchanged under application of the intermediately operator.

Properties obtained via this operator

Naturally arising properties that satisfy intermediate subgroup condition

These include:

• Normal subgroup
• Central factor
• Conjugacy-closed subgroup
• Permutable subgroup
• Modular subgroup

Subgroup properties obtained by explicitly applying the intermediately operator

• Intermediately characteristic subgroup
• Compressed subgroup