Transitive subgroup property

This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
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This article is about a general term. A list of important particular cases (instances) is available at Category:Transitive subgroup properties

Origin

Origin of the concept

The fact that normal subgroups of normal subgroups need not be normal dates back to the beginnings of group theory, and translation of this to statements like normality is not transitive indicates that the notion of transitivity was there.

Origin of the term

The notion of transitive relation is basic to the theory of relations. A transitive property is essentially a subgroup property which when loosely viewed as a relation, is transitive (it cannot strictly be viewed as an relation because an embedding of a subgroup in a group is more than merely a relation between their isomorphism classes).

However, its formal use is only in this wiki.

Definition

Symbol-free definition

A subgroup property is said to be transitive if the composite of two subgroup embeddings, both having the property, also must have the property.

Definition with symbols

A subgroup property $p$ is termed transitive if, whenever $G$ satisfies $p$ as a subgroup of $H$, and $H$ satisfies the property as a subgroup of $K$, then $G$ satisfies $p$ as a subgroup of $K$.

Definition in terms of the composition operator

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

Property theory of transitive subgroup properties

Related metaproperties

A metaproperty closely related to transitivity is that of being identity-true. This is the property of a subgroup property being weaker than the improper property (which is the identity element for the composition operator). A subgroup property that is both transitive and identity-true is termed trim.

Remedies for lack of transitivity

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 transitive identity-true properties.

• The left transiter, which takes a subgroup property $p$ and returns the maximum subgroup property $q$ such that $q*p$ ≤ $p$. The left transiter operator is an idempotent operator, but it is in general neither ascendant nor monotone. It is descendant when applied to an identity-true subgroup property.

• The right transiter, which takes a subgroup property $p$ and returns the maximum subgroup property $q$ such that $p*q$ ≤ $p$. The left transiter operator is an idempotent operator, but it is in general neither ascendant nor monotone. It is descendant when applied to an identity-true subgroup property.

• The subordination operator, which is a Kleene star closure for the subgroup property. This operator is idempotent, ascendant, and monotone.

Naturally arising transitive subgroup properties

Because of balancedness

Further information: Balanced subgroup property

Many subgroup properties that arise naturally are transitive. For instance, any balanced subgroup property with respect to a restriction formalism or an extension formalism is transitive.

Here's a list of important subgroup properties that are balanced, cateogrized by the reason:

• Because of being balanced with respect to the function restriction formalism: The properties of being characteristic, fully characteristic, transitively normal, a central factor, a conjugacy-closed normal subgroup.
• Because of being balanced with respect to the function extension formalism: The properties of being an AEP subgroup, an EEP subgroup.
• Because of being balanced with respect to the representation restriction formalism: The properties of being a conjugacy-closed normal subgroup, and of being a subgroup to which every irreducible representation of the whole group restricts to an irreducible representation.
• Because of being balanced with respect to the equivalence relation restriction formalism: The property of being a conjugacy-closed subgroup
• Because of being balanced with respect to the subgroup intersection restriction formalism: The property of being a maximal-sensitive subgroup.
• Because of being balanced with respect to the the subgroup intersection extension formalism: The property of being a CEP subgroup.

Because of a suitable composition/multiplication

The proofs for each of these properties is somewhat ad hoc:

• The property of being a direct factor: Here, we take the product of the normal complements for the two subgroup embeddings. Equivalently, we compose the two projection mappings.
• The property of being a retract: Here, we take the products of the normal complements for the two subgroup embeddings. Equivalently, we compose the two retraction mappings.

Because the subgroup property is an equational/expressional closure

The proofs basically rely on iterating the fact that things in the main group imply things in the subgroup. These can be cast in terms of restriction formalisms, but the formalism exercise may not be worthwhile. Examples:

• The property of being a serving subgroup
• The property of being an equation-closed subgroup.
• The property of being a verbal subgroup.
• The property of being a existentially bound-word subgroup.
• The property of being a malnormal subgroup.

Because the subgroup property is left hereditary or right hereditary

Any left hereditary property is clearly transitive, as is any right hereditary property. Thus for instance the property of being hereditarily normal is transitive.

Properties that are not transitive

Antitransitive properties

A subgroup property $p$ is termed antitransitive if whenever $H$ < $K$ $G$ with inclusions strict, $H$ satisfies $p$ in $K$ and $K$ satisfies $p$ in $G$, then $H$ cannot satisfy $p$ in $G$.

Any left antihereditary property is antitransitive, and so is any right antihereditary property. In particular, minimal and maximal properties are antitransitive.

A property is both transitive and antitransitive if and only if the square of its conjunction with properness is the fallacy.

Some examples of antitransitive properties that are not transitive: the property of being minimal normal, the property of being maximal normal.

Non-balanced properties

Properties that can be expressed using a restriction formalism but for which the expression cannot be balanced, are not transitive. The classical example is the property of normality, where right tightening leaves the right side at automorphisms, while the left side is at inner automorphisms.