Antitransitive subgroup property

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

This is an opposite of transitive subgroup property

Definition

Symbol-free definition

A subgroup property is said to be antitransitive if its conjunction with the square of its proper part is the fallacy.

Definition with symbols

A subgroup property $p$ is said to be antitransitive if whenever $H<K<G$ are groups with all the inclusions strict, such that $H$ satisfies $p$ in $K$ and $K$ satisfies $p$ in $G$, then $H$ cannot satisfy $p$ in $G$.

Relation with other metaproperties

Stronger metaproperties

• Left-antihereditary subgroup property
• Right-antihereditary subgroup property

Relation with transitivity

A subgroup property can be both transitive and antitransitive if and only if the square of its proper part is the fallacy, or equivalently, its proper part is a nil squareroot.