Core-characteristic subgroup: Difference between revisions
No edit summary |
|||
| (6 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
{{wikilocal}} | |||
{{subgroup property}} | {{subgroup property}} | ||
==Definition== | |||
{{quick phrase|[[quick phrase::intersection of all conjugates is characteristic]], [[quick phrase::normal core is characteristic]]}} | |||
===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[subgroup]] of a [[group]] is termed '''core-characteristic''' if its [[normal core]] is a [[characteristic subgroup]] of the whole group. | A [[subgroup]] of a [[group]] is termed '''core-characteristic''' if its [[defining ingredient::normal core]] is a [[defining ingredient::characteristic subgroup]] of the whole group. | ||
===Definition with symbols=== | ===Definition with symbols=== | ||
| Line 17: | Line 18: | ||
===Stronger properties=== | ===Stronger properties=== | ||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::characteristic subgroup]] || invariant under all [[automorphism]]s|| || || {{intermediate notions short|core-characteristic subgroup|characteristic subgroup}} | |||
|- | |||
| [[Weaker than::automorph-dominating subgroup]] || every [[automorphic subgroups|automorphic subgroup]] is contained in a [[conjugate subgroups|conjugate subgroup]] || || || {{intermediate notions short|core-characteristic subgroup|automorph-dominating subgroup}} | |||
|- | |||
| [[Weaker than::automorph-conjugate subgroup]] || every [[automorphic subgroups|automorphic subgroup]] is [[conjugate subgroups|conjugate]] to it || (via automorph-dominating) || || {{intermediate notions short|core-characteristic subgroup|automorph-conjugate subgroup}} | |||
|- | |||
| [[Weaker than::intersection of automorph-conjugate subgroups]] || intersection of [[automorph-conjugate subgroup]]s || || || {{intermediate notions short|core-characteristic subgroup|intersection of automorph-conjugate subgroups}} | |||
|- | |||
| [[Weaker than::core-free subgroup]] || [[normal core]] is trivial || || || {{intermediate notions short|core-characteristic subgroup|core-free subgroup}} | |||
|- | |||
| [[Weaker than::Sylow subgroup]] || <math>p</math>-subgroup of finite group whose index is relatively prime to <math>p</math> || || || {{intermediate notions short|core-characteristic subgroup|Sylow subgroup}} | |||
|- | |||
| [[Weaker than::Hall subgroup]] || subgroup of finite group whose order and index are relatively prime || || || {{intermediate notions short|core-characteristic subgroup|Hall subgroup}} | |||
|} | |||
===Conjunction with other properties=== | ===Conjunction with other properties=== | ||
Latest revision as of 17:34, 21 December 2014
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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 | |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) | |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 | |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.