Transitive subgroup property

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
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History

This term is local to the wiki. To learn more about why this name was chosen for the term, and how it does not conflict with existing choice of terminology, refer the talk page

Definition

A subgroup property $p$ is termed transitive if, whenever $H \leq K \leq G$ such that:

$H$ satisfies $p$ as a subgroup of $K$ , and
$K$ satisfies $p$ as a subgroup of $G$ ,

Then $H$ satisfies $p$ as a subgroup of $G$ .

Definition in terms of the composition operator

If $*$ is the composition operator on subgroup properties, then a property $p$ is transitive if $p * p \leq p$ .

Examples

The examples chosen below are a mix of important properties and other diverse properties to give a sense of how transitivity can play out in different flavors.

For a wider range of examples organized by theme, see proving transitivity.

Subgroup property	Full proof of transitivity	Proof idea
characteristic subgroup	characteristicity is transitive	We can write characteristicity as the balanced subgroup property with respect to the function property of being an automorphism, i.e., a subgroup of a group is characteristic if and only if every automorphism of the whole group restricts to an automorphism of the subgroup. Balanced implies transitive, so this suffices.
fully invariant subgroup	full invariance is transitive	We can write full invariance as the balanced subgroup property with respect to the function property of being an endomorphism, i.e., a subgroup of a group is fully invariant if and only if every endomorphism of the whole group restricts to an endomorphism of the subgroup. Balanced implies transitive, so this suffices.
powering-invariant subgroup	powering-invariance is transitive	We can write powering-invariance as the balanced subgroup property with respect to the function property of being an rational power map, i.e., a subgroup of a group is powering-invariant if and only if every rational power map of the whole group restricts to a rational power map of the subgroup. Balanced implies transitive, so this suffices.
subgroup of finite index	index is multiplicative	The proof uses a map of coset spaces whose fibers are the third coset space.
automorph-conjugate subgroup	automorph-conjugacy is transitive	The proof is basically element-chasing, and can be obtained by following your nose.
Hall subgroup	Hall is transitive	A Hall subgroup is a subgroup whose order and index are relatively prime. The proof is straightforward from basic number theory.
direct factor	direct factor is transitive	This follows from the associativity of direct products.
retract	retract is transitive	This follows from the fact that the composite of two retractions is a retraction.

Relation with other metaproperties

Stronger metaproperties

Metaproperty	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
t.i. subgroup property	This is the conjunction of being transitive with being identity-true: in other words, a subgroup property is t.i. if it is transitive, and is always satisfied by any group as a subgroup of itself.
balanced subgroup property	These are special kinds of transitive subgroup properties that can be expressed via function restriction expressions with both sides being the same.
idempotent subgroup property	This is a subgroup property $p$ such that $p * p = p$ .
Left-hereditary subgroup property	Any subgroup of a subgroup with this property has this property.

Opposite metaproperties

Antitransitive subgroup property

Metametaproperties

Metametaproperty name	Satisfied?	Proof	Statement with symbols
conjunction-closed subgroup metaproperty	Yes		The conjunction (AND) of a collection of subgroup properties, each of which is transitive, is also transitive. A corollary of this is that given any subgroup property, there is a strongest transitive subgroup property implied by it; that subgroup property is given by the subordination operator.
disjunction-closed subgroup metaproperty	No	transitivity is not disjunction-closed	The disjunction (OR) of two transitive subgroup properties need not be transitive. In other words, it is possible to find subgroup properties $p_{1}, p_{2}$ that are both transitive but such that the property $p_{1} \lor p_{2}$ is not transitive.

Operators to make a subgroup property transitive

There are three general ways to pass from a general subgroup property to a transitive variation (The term variation could be misleading, as we shall see). Each of these is an idempotent operator and the fixed point space is precisely the space of t.i. subgroup properties. These are:

Operator	Description
left transiter	operator takes a subgroup property $p$ and returns the maximum subgroup property $q$ such that $q * p \leq p$ . In other words, $H$ satisfies property $q$ in $K$ if whenever $K$ has property $p$ in $G$ , $H$ also has property $p$ in $G$ . It turns out, from the transiter master theorem, that the left transiter of any subgroup property is a t.i. subgroup property (that is, both transitive and identity-true), and further, that the left transiter of a t.i. subgroup property is itself.
right transiter	operator takes a subgroup property $p$ and returns the maximum subgroup property $q$ such that $p * q \leq p$ . In other words, $K$ satisfies property $q$ in $G$ if and only if for every subgroup $H$ satisfying $p$ in $K$ , $H$ must also satisfy $p$ in $G$ .
subordination operator (we can think of this as the Kleene star closure with respect to the composition operator)	The subordination of a property $p$ is the property of being a subgroup such that there is a finite length chain from the subgroup to the whole group wherein each has property $p$ in its successor. Note that we allow a chain of length zero.

Other ways are:

Effect of property modifiers on transitivity

Transfer condition operator

Further information: Transfer condition operator preserves transitivity

Let $p$ be a subgroup property. The transfer-closure of $p$ is defined as the following subgroup property $q$ : A subgroup $H$ has property $q$ in $G$ if $H \cap K$ has property $p$ in $K$ for any subgroup $K$ of $G$ that has property $p$ .

Then, if $p$ is transitive, so is the transfer-closure of $p$ .

Intermediately operator

The intermediately operator may not in general preserve transitivity.

Testing

GAP code

One can write code to test this subgroup metaproperty in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this subgroup metaproperty with given biggest group at: IsTransitiveWithBigGroupView the GAP code for testing this subgroup metaproperty on all groups living inside a given big group at: IsTransitiveInAllSubgroupsOfGroup
View other GAP-codable subgroup metaproperties

It is possible to check, given a group $G$ and a subgroup property $p$ , whether whenever $H \leq K \leq G$ are subgroups such that $H$ satisfies property $p$ in $K$ and $K$ satisfies property $p$ in $G$ , $H$ also satisfies property $p$ in $G$ . Although there is no in-built command for this, it can be achieved using a short snippet of code, available at GAP:IsTransitiveWithBigGroup. This is then used as follows:

IsTransitiveWithBigGroup(Group,Property)

We can also check whether, for a given property $p$ and a group $G$ , whenever $H \leq K \leq L \leq G$ are such that $H$ satisfies property $p$ in $K$ and $K$ satisfies property $p$ in $L$ , then $H$ satisfies property $p$ in $L$ . The short snippet of code needed for this is available at GAP:IsTransitiveInAllSubgroupsOfGroup. It is used as follows:

IsTransitiveInAllSubgroupsOfGroup(Group,Property)