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: In particular, any normal Sylow subgroup is characteristic. Thus, this subgroup property is normal-to-characteristic

Weaker properties

• Hall subgroup
• Order-dominating subgroup
• Isomorph-conjugate subgroup
• Automorph-conjugate subgroup
• order-automorphic subgroup
• Isomorph-automorphic subgroup
• Intermediately automorph-conjugate subgroup
• Pronormal subgroup
• Closure-characteristic subgroup
• Core-characteristic subgroup

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
• Domination(D): Any $p$-group is contained in a $p$-Sylow subgroup
• Conjugacy(C): Any two $p$-Sylow subgroups are conjugate

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

Transfer condition

YES: This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.
View other subgroup properties satisfying the transfer condition

If $H$ is a Sylow subgroup of $G$, and $K$ is any subgroup, then the intersection of $H$ and $K$ is a Sylow subgroup of $K$.

References

Textbook references

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