Join of automorph-conjugate subgroups: Difference between revisions

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==Definition==
==Definition==


A [[subgroup]] of a [[group]] is said to be '''AC-generated''' if it is generated by a collection of [[automorph-conjugate subgroup]]s, viz it is a join of automorph-conjugate subgroups.
A [[subgroup]] of a [[group]] is termed a '''join of automorph-conjugate subgroups''' or '''AC-generated''' if it is generated by a collection of [[automorph-conjugate subgroup]]s, viz it is a join of automorph-conjugate subgroups.


==Facts==
==Relation with other properties==


Any normal AC-generated subgroup is [[characteristic subgroup|characteristic]]. Thus, being AC-generated is a [[normal-to-characteristic subgroup property]].
===Stronger properties===
 
{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Weaker than::automorph-conjugate subgroup]] || [[conjugate subgroups|conjugate]] to all its [[automorphic subgroups]] || (obvious) || [[automorph-conjugacy is not finite-join-closed]] || {{intermediate notions short|join of automorph-conjugate subgroups|automorph-conjugate subgroup}}
|-
| [[Weaker than::join of Sylow subgroups]] || join of [[Sylow subgroup]]s || (obvious) || Any proper nontrivial characteristic subgroup of a <math>p</math>-group, e.g., [[Z2 in Z4]] || {{intermediate notions short|join of automorph-conjugate subgroups|join of Sylow subgroups}}
|-
| [[Weaker than::Hall subgroup]] || subgroup whose order and index are relatively prime || ([[Hall implies join of Sylow subgroups|via join of Sylow subgroups]]) || (via join of Sylow subgroups) || {{intermediate notions short|join of automorph-conjugate subgroups|Hall subgroups}}
|}
 
===Weaker properties===
 
{| class="sortable" border="1"
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
|-
| [[Stronger than::closure-characteristic subgroup]] || its [[normal closure]] is a [[characteristic subgroup]] || || || {{intermediate notions short|closure-characteristic subgroup|join of automorph-conjugate subgroups}}
|-
| [[Stronger than::normal-to-characteristic subgroup]] || if normal, it is also characteristic || || || {{intermediate notions short|normal-to-characteristic subgroup|join of automorph-conjugate subgroups}}
|}
 
===Conjunction with other properties===
 
Any normal subgroup that is a join of automorph-conjugate subgroups is [[characteristic subgroup|characteristic]]. {{normal-to-characteristic}}

Latest revision as of 21:17, 20 December 2014

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]

Definition

A subgroup of a group is termed a join of automorph-conjugate subgroups or AC-generated if it is generated by a collection of automorph-conjugate subgroups, viz it is a join of automorph-conjugate subgroups.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
automorph-conjugate subgroup conjugate to all its automorphic subgroups (obvious) automorph-conjugacy is not finite-join-closed |FULL LIST, MORE INFO
join of Sylow subgroups join of Sylow subgroups (obvious) Any proper nontrivial characteristic subgroup of a p-group, e.g., Z2 in Z4 |FULL LIST, MORE INFO
Hall subgroup subgroup whose order and index are relatively prime (via join of Sylow subgroups) (via join of Sylow subgroups) |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
closure-characteristic subgroup its normal closure is a characteristic subgroup |FULL LIST, MORE INFO
normal-to-characteristic subgroup if normal, it is also characteristic |FULL LIST, MORE INFO

Conjunction with other properties

Any normal subgroup that is a join of automorph-conjugate subgroups is characteristic. Thus, this subgroup property is normal-to-characteristic

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