Sylow subgroup

This article is about a standard (though not very rudimentary) definition in group theory. The article text may, however, contain more than just the basic definition
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The article defines a subgroup property, where the definition may be in terms of a particular prime number that serves as parameter
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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
• Abelian Sylow subgroup

Weaker properties

• Order-dominating subgroup: For full proof, refer: Sylow implies order-dominating Also related:
  • Order-dominated subgroup: For full proof, refer: Sylow implies order-dominated
  • 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
• Hall subgroup
  • Nilpotent Hall subgroup
  • Order-dominating Hall subgroup
• Procharacteristic subgroup: For full proof, refer: Sylow implies procharacteristic Also related:
  • Weakly procharacteristic subgroup
  • Paracharacteristic subgroup
  • Polycharacteristic subgroup
  • Weakly characteristic subgroup
  • Intermediately normal-to-characteristic subgroup
• Pronormal subgroup: For full proof, refer: Sylow implies pronormal Also related:
  • Weakly pronormal subgroup
  • Paranormal subgroup
  • Polynormal subgroup
  • Weakly normal subgroup
• Intermediately subnormal-to-normal subgroup
• Subgroup with abnormal normalizer: Also related:
  • Subgroup with weakly abnormal normalizer
  • Subgroup with self-normalizing normalizer
• Closure-characteristic subgroup
• Core-characteristic subgroup
• WNSCDIN-subgroup: For full proof, refer: Sylow implies WNSCDIN Also related:
  • MWNSCDIN-subgroup: For full proof, refer: Sylow implies MWNSCDIN
  • Subgroup in which every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed: For full proof, refer: WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed
• Intermediately normal-to-complemented subgroup: Also related:
  • Intermediately central factor-to-direct factor
  • Intermediately conjugacy-closed-to-retract: For full proof, refer: Conjugacy-closed and Sylow implies retract

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 $H,K\leq G$, with $H$ a Sylow subgroup of $G$, $H\cap K$ need not be a Sylow subgroup of $K$. However, if $HK=KH$, then $H\cap K$ is a Sylow subgroup of $K$.For full proof, refer: Sylow does not satisfy transfer condition, Sylow satisfies permuting 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)