Core-characteristic subgroup
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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.
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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 H of a group G is termed core-characteristic if the normal core HG of H in G is a characteristic subgroup of G.
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 | click here | ||
| Automorph-conjugate subgroup | every automorphic subgroup is conjugate to it | click here | ||
| Intersection of automorph-conjugate subgroups | intersection of automorph-conjugate subgroups | |||
| Core-free subgroup | normal core is trivial | |||
| Sylow subgroup | p-subgroup of finite group whose index is relatively prime to p | click here | ||
| Hall subgroup | subgroup of finite group whose order and index are relatively prime |
Conjunction with other properties
Any normal subgroup that is also core-characteristic, is characteristic.
Incomparable properties
Facts about Core-characteristic subgroupRDF feed
| Defining ingredient | Normal core +, and Characteristic subgroup + |
| Page class | Term + |
| Quick phrase | intersection of all conjugates is characteristic +, and normal core is characteristic + |
| Weaker than | Characteristic subgroup +, Automorph-conjugate subgroup +, Intersection of automorph-conjugate subgroups +, Core-free subgroup +, Sylow subgroup +, and Hall subgroup + |
