Transfer condition
From Groupprops
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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
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 is said to satisfy the transfer condition if whenever
satisfies property
as a subgroup of
, and
is a subgroup of
, then
∩
satisfies property
as a subgroup of
.
Formalisms
This article defines a single-input-expressible subgroup metaproperty
Consider the procedure that takes as input a group-subgroup pair
, and outputs all group-subgroup pairs
for
. The transfer condition is the single-input-expressible metaproperty corresponding to procedure
: a subgroup property
satisfies the transfer condition if
satisfying property
implies that all pairs
also satisfy property
.
Relation with other metaproperties
Stronger metaproperties
Weaker metaproperties
Conjunction implications
- Any transitive subgroup property that satisfies the transfer condition is also finite-intersection-closed. For full proof, refer: Transitive and transfer condition implies finite-intersection-closed
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