Sylow subgroup

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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 it is a $p$-Sylow subgroup for some prime number $p$. In other words, it satisfies the following equivalent conditions:

1. (Order-index definition): It is a $p$-group (i.e., its order is a power of $p$, so it is a group of prime power order) and its index is relatively prime to $p$.
2. (Maximal power of prime definition): There is a prime $p$ such that the order of the subgroup is the largest power of $p$ dividing the order of the group.
3. (Hall subgroup plus p-subgroup definition): It is a $p$-group and also a Hall subgroup: its order and index in the whole group are relatively prime.

Note that the trivial subgroup is always a Sylow subgroup: it is $p$-Sylow for any prime $p$ not dividing the order of the group. The whole group is $p$-Sylow as a subgroup of itself if and only if it is a $p$-group.

Definition with symbols

A subgroup $P$ of a finite group $G$ is termed a Sylow subgroup if it is a $p$-Sylow subgroup for some prime number $p$. In other words, it satisfies the following equivalent conditions:

1. (Order-index definition): $P$ is a $p$-group (i.e., its order is $p^r$ for some nonnegative integer $r$), so it is a group of prime power order) and its index $[G:P]$ is relatively prime to $p$.
2. (Maximal power of prime definition): There is a prime $p$ such that the order of $P$ is the largest power of $p$ dividing the order of $G$. In other words, if the order of $G$ is $p^{r}m$ where $r$ is a nonnegative integer and $m$ is an integer relatively prime to $p$, $P$ must have order $p^r$.
3. (Hall subgroup plus p-subgroup definition): It is a $p$-group and also a Hall subgroup: its order $\left|P\right|$ and index $[G:P]$ are relatively prime.

Relation with other properties

Conjunction with other properties

• Normal Sylow subgroup
• Sylow direct factor
• Sylow retract
• Abelian Sylow subgroup

Weaker properties

For a better understanding of how all these facts about Sylow subgroups are proved, refer the survey articles deducing basic facts about Sylow subgroups and Hall subgroups and deducing advanced facts about Sylow subgroups and Hall subgroups

• 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 $H,K\le G$, 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)