Join of Sylow subgroups: Difference between revisions
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==Definition== | |||
A subgroup <math>H</math> of a [[finite group]] <math>G</math> is termed a '''join of Sylow subgroups''' if it can be expressed as a [[join of subgroups|join]] of [[Sylow subgroup]]s of <math>G</math>. There are no restrictions on the [[prime number]]s we can use for the Sylow subgroups: we could use a join of Sylow subgroups all for the same prime, or all for different primes, or with multiple primes, some of which are used multiple times. | |||
==Formalisms== | ==Formalisms== | ||
{{obtainedbyapplyingthe|join operator|Sylow subgroup}} | {{obtainedbyapplyingthe|join operator|Sylow subgroup}} | ||
==Relation with other properties== | |||
===Stronger properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Weaker than::Sylow subgroup]] || subgroup of maximal prime power order in finite group || (obvious) || (obvious) || {{intermediate notions short|join of Sylow subgroups|Sylow subgroup}} | |||
|- | |||
| [[Weaker than::Hall subgroup]] || subgroup whose order and index are relatively prime || [[Hall implies join of Sylow subgroups]] || [[join of Sylow subgroups not implies Hall]] || {{intermediate notions short|join of Sylow subgroups|Hall subgroup}} | |||
|- | |||
| [[Weaker than::Sylow-closure]] || [[normal closure]] of a Sylow subgroup || join of all the conjugates of that Sylow subgroup || || {{intermediate notions short|join of Sylow subgroups|Sylow-closure}} | |||
|} | |||
===Weaker properties=== | |||
{| class="sortable" border="1" | |||
! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | |||
|- | |||
| [[Stronger than::join of automorph-conjugate subgroups]] || join of subgroups of the group each of which is an [[automorph-conjugate subgroup]] of the whole group || follows from [[Sylow implies automorph-conjugate]] || || {{intermediate notions short|join of automorph-conjugate subgroups|join of Sylow subgroups}} | |||
|- | |||
| [[Stronger than::closure-characteristic subgroup]] || [[normal closure]] is a [[characteristic subgroup]] || (via join of automorph-conjugate subgroups) || || {{intermediate notions short|closure-characteristic subgroup|join of Sylow subgroups}} | |||
|} | |||
Revision as of 19:11, 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]
Definition
A subgroup of a finite group is termed a join of Sylow subgroups if it can be expressed as a join of Sylow subgroups of . There are no restrictions on the prime numbers we can use for the Sylow subgroups: we could use a join of Sylow subgroups all for the same prime, or all for different primes, or with multiple primes, some of which are used multiple times.
Formalisms
In terms of the join operator
This property is obtained by applying the join operator to the property: Sylow subgroup
View other properties obtained by applying the join operator
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Sylow subgroup | subgroup of maximal prime power order in finite group | (obvious) | (obvious) | |FULL LIST, MORE INFO |
| Hall subgroup | subgroup whose order and index are relatively prime | Hall implies join of Sylow subgroups | join of Sylow subgroups not implies Hall | |FULL LIST, MORE INFO |
| Sylow-closure | normal closure of a Sylow subgroup | join of all the conjugates of that Sylow subgroup | |FULL LIST, MORE INFO |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| join of automorph-conjugate subgroups | join of subgroups of the group each of which is an automorph-conjugate subgroup of the whole group | follows from Sylow implies automorph-conjugate | |FULL LIST, MORE INFO | |
| closure-characteristic subgroup | normal closure is a characteristic subgroup | (via join of automorph-conjugate subgroups) | |FULL LIST, MORE INFO |