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]
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
QUICK PHRASES: normal closure is characteristic, join of all conjugates is characteristic
A subgroup of a group is termed closure-characteristic, or a subgroup whose normal closure is characteristic, if its normal closure in the whole group is a characteristic subgroup. In symbols, a subgroup of a group is termed closure-characteristic if the normal closure of in is characteristic in .
Relation with other properties
Conjunction with other properties
Any normal subgroup that is also closure-characteristic, is characteristic.
BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)
|Metaproperty name||Satisfied?||Proof||Difficulty level (0-5)||Statement with symbols|
|trim subgroup property||Yes||(obvious)||0||In any group , both the trivial subgroup and the whole group are closure-characteristic.|
|strongly join-closed subgroup property||Yes||closure-characteristicity is strongly join-closed||Suppose is a group and are all closure-characteristic subgroups of . Then the join is also closure-characteristic.|