Transfer condition

From Groupprops
Jump to: navigation, search
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
This article is about a general term. A list of important particular cases (instances) is available at Category:Subgroup properties satisfying transfer condition

Definition

Definition with symbols

A subgroup property p is said to satisfy the transfer condition if whenever H satisfies property p as a subgroup of G, and K is a subgroup of G, then HK satisfies property p as a subgroup of K.

Formalisms

This article defines a single-input-expressible subgroup metaproperty

Consider the procedure P that takes as input a group-subgroup pair H \le G, and outputs all group-subgroup pairs H \cap K \le K for K \le G. The transfer condition is the single-input-expressible metaproperty corresponding to procedure P: a subgroup property p satisfies the transfer condition if H \le G satisfying property p implies that all pairs H \cap K \le K also satisfy property p.

Relation with other metaproperties

Stronger metaproperties

Weaker metaproperties

Conjunction implications

Metametaproperties

Conjunction-closedness

This subgroup metaproperty is conjunction-closed: an arbitrary conjunction (AND) of subgroup properties satisfying this metaproperty, also satisfies this metaproperty
View conjunction-closed subgroup metaproperties

Disjunction-closedness

This subgroup metaproperty is disjunction-closed: an arbitrary disjunction (OR) of subgroup properties satisfying this metaproperty, also satisfies this metaproperty
View all disjunction-closed subgroup metaproperties

Composition-closedness

This subgroup metaproperty is composition-closed: the property obtained by applying the composition operator to two subgroup properties satisfying this metaproperty, also satisfies this metaproperty
View a complete list of composition-closed subgroup metaproperties