Transfer-closed characteristic subgroup
From Groupprops
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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]
This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity
Definition
Definition with symbols
A subgroup of a group
is termed a transfer-closed characteristic subgroup if, for any subgroup
,
is a characteristic subgroup of
.
Formalisms
In terms of the transfer condition operator
This property is obtained by applying the transfer condition operator to the property: characteristic subgroup
View other properties obtained by applying the transfer condition operator
Relation with other properties
Stronger properties
- Normal Sylow subgroup
- Normal Hall subgroup
- Variety-containing subgroup
- Subisomorph-containing subgroup
Weaker properties
Metaproperties
Metaproperty name | Satisfied? | Proof | Difficulty level | Statement with symbols |
---|---|---|---|---|
transitive subgroup property | Yes | transfer-closed characteristicity is transitive | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
intermediate subgroup condition | Yes | If ![]() ![]() ![]() ![]() ![]() | ||
strongly intersection-closed subgroup property | Yes | If ![]() ![]() ![]() ![]() | ||
transfer condition | Yes | If ![]() ![]() ![]() ![]() ![]() ![]() ![]() |