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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]
Definition
QUICK PHRASES: intersection of all conjugates is characteristic, normal core is characteristic
Symbol-free definition
A subgroup of a group is termed core-characteristic if its normal core is a characteristic subgroup of the whole group.
Definition with symbols
A subgroup
of a group
is termed core-characteristic if the normal core
of
in
is a characteristic subgroup of
.
Relation with other properties
Stronger properties
Property |
Meaning |
Proof of implication |
Proof of strictness (reverse implication failure) |
Intermediate notions
|
characteristic subgroup |
invariant under all automorphisms |
|
|
Automorph-conjugate subgroup, Intersection of automorph-conjugate subgroups, Intersection-transitively automorph-conjugate subgroup, Join-transitively automorph-conjugate subgroup|FULL LIST, MORE INFO
|
automorph-dominating subgroup |
every automorphic subgroup is contained in a conjugate subgroup |
|
|
|FULL LIST, MORE INFO
|
automorph-conjugate subgroup |
every automorphic subgroup is conjugate to it |
(via automorph-dominating) |
|
Automorph-dominating subgroup, Intersection of automorph-conjugate subgroups|FULL LIST, MORE INFO
|
intersection of automorph-conjugate subgroups |
intersection of automorph-conjugate subgroups |
|
|
|FULL LIST, MORE INFO
|
core-free subgroup |
normal core is trivial |
|
|
|FULL LIST, MORE INFO
|
Sylow subgroup |
-subgroup of finite group whose index is relatively prime to |
|
|
Automorph-conjugate subgroup, Hall subgroup, Intersection of Sylow subgroups, Subgroup whose normal core is fully invariant|FULL LIST, MORE INFO
|
Hall subgroup |
subgroup of finite group whose order and index are relatively prime |
|
|
|FULL LIST, MORE INFO
|
Conjunction with other properties
Any normal subgroup that is also core-characteristic, is characteristic.
Incomparable properties