Sylow subgroup

The article defines a subgroup property, where the definition may be in terms of a particular prime number that serves as parameter
View other prime-parametrized subgroup properties | View all subgroup properties

This article describes a property that arises as the conjunction of a subgroup property: Hall subgroup with a group property (itself viewed as a subgroup property): group of prime power order
View a complete list of such conjunctions

Definition

Symbol-free definition

A subgroup of a finite group is termed a Sylow subgroup if there is a prime $p$ for which it is a $p$-group and its index is relatively prime to $p$, or equivalently, if it is a $p$-group and also a Hall subgroup.

Relation with other properties

Conjunction with other properties

• Normal Sylow subgroup
• Sylow direct factor
• Sylow retract

Weaker properties

• Nilpotent Hall subgroup
• Hall subgroup
• Order-dominating subgroup: For full proof, refer: Sylow implies order-dominating
• Order-conjugate subgroup: For full proof, refer: Sylow implies order-conjugate
• Order-automorphic subgroup
• Order-isomorphic subgroup
• Isomorph-automorphic subgroup: For full proof, refer: Sylow implies isomorph-automorphic
• Isomorph-conjugate subgroup: For full proof, refer: Sylow implies isomorph-conjugate
• Automorph-conjugate subgroup: For full proof, refer: Sylow implies automorph-conjugate
• Intermediately isomorph-conjugate subgroup: For full proof, refer: Sylow implies intermediately isomorph-conjugate
• Intermediately automorph-conjugate subgroup: For full proof, refer: Sylow implies intermediately automorph-conjugate
• Procharacteristic subgroup: For full proof, refer: Sylow implies procharacteristic
• Pronormal subgroup: For full proof, refer: Sylow implies pronormal
• Weakly procharacteristic subgroup
• Weakly pronormal subgroup
• Paracharacteristic subgroup
• Paranormal subgroup
• Polycharacteristic subgroup
• Polynormal subgroup
• Subgroup with abnormal normalizer
• Closure-characteristic subgroup
• Core-characteristic subgroup
• Intermediately normal-to-characteristic subgroup
• Intermediately subnormal-to-normal subgroup
• WNSCDIN-subgroup: For full proof, refer: Sylow implies WNSCDIN
• Subgroup in which every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed

Metaproperties

No proper nontrivial subgroup of a Sylow subgroup can be a Sylow subgroup.

ECD

The property of being a $p$-Sylow subgroup is obtained as the property of being maximal corresponding to the group property of being a $p$-Sylow subgroup. It turns out that:

• Existence (E): For every $p$, there exit $p$-Sylow subgroups. For full proof, refer: Sylow subgroups exist
• Domination(D): Any $p$-group is contained in a $p$-Sylow subgroup. For full proof, refer: Sylow implies order-dominating
• Conjugacy(C): Any two $p$-Sylow subgroups are conjugate. For full proof, refer: Sylow implies order-conjugate

All these facts, together, show that the group property of being a $p$-group satisfies the ECD condition.

Intermediate subgroup condition

YES: This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.
ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup condition
ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If $H$ is a Sylow subgroup of $G$, and $K$ is any intermediate subgroup of $G$ containing $H$, then $H$ is a Sylow subgroup of $K$. For full proof, refer: Sylow satisfies intermediate subgroup condition

Transfer condition

This subgroup property does not satisfy the transfer condition

If $H,K\le G$, with $H$ a Sylow subgroup of $G$, $H\cap K$ need not be a Sylow subgroup of $K$. For full proof, refer: Sylow does not satisfy transfer condition

Image condition

YES: This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property
View other subgroup properties satisfying image condition

For full proof, refer: Sylow satisfies image condition

References

Textbook references

• Algebra by Michael Artin, ISBN 0130047635, 13-digit ISBN 978-0130047632, ^{More info}, Page 206, Point (4.5) (formal definition)