Transitive subgroup property

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BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions

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 \le K \le 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 \le p.

Related survey articles

The following survey articles discuss transitivity:

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

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 Yes 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 \le 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 \le 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.

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 \le K \le 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 \le K \le L \le 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)