Sylow subgroup
Definition
Equivalent definitions in tabular format
A subgroup of a finite group is termed a Sylow subgroup if it is a -Sylow subgroup for some prime number . We give equivalent definitions of a -Sylow subgroup.
No. | Shorthand | A subgroup of a group is a -Sylow subgroup if... | A subgroup of a group is a -Sylow subgroup of if ... | Applications to... | Additional comments |
---|---|---|---|---|---|
1 | order prime, index coprime | it is a -group (i.e., its order is a power of , so it is a group of prime power order) and its index is relatively prime to | is a -group (i.e., its order is for some nonnegative integer ), so it is a group of prime power order) and its index is relatively prime to . | ||
2 | order is maximal power of prime definition | the order of the subgroup is the largest power of dividing the order of the group | the order of is the largest power of dividing the order of . In other words, if the order of is where is a nonnegative integer and is an integer relatively prime to , must have order . | ||
3 | Hall subgroup plus p-subgroup definition | it is a -group (nontrivial if divides the group order) and also a Hall subgroup: its order and index in the whole group are relatively prime | it is a -group (nontrivial if divides the order of ) and also a Hall subgroup: its order and index are relatively prime. | ||
4 | maximal element among p-subgroups | it is a -subgroup of the whole group and is not contained in any bigger -subgroup | is a -subgroup of (i.e., the order of is a power of ) and any -subgroup of containing must equal |
Note that the trivial subgroup is always a Sylow subgroup: it is -Sylow for any prime not dividing the order of the group. The whole group is -Sylow as a subgroup of itself if and only if it is a -group.
Equivalence of definitions
Note that the equivalence of definitions (1)-(3) is a simple exercise in Lagrange's theorem and basic facts about prime factorization. The equivalence with definition (4) relies on the fact that Sylow subgroups exist and Sylow implies order-dominating, which are both considered parts of Sylow's theorem.
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
VIEW: Definitions built on this | Facts about this: (facts closely related to Sylow subgroup, all facts related to Sylow subgroup) |Survey articles about this | Survey articles about definitions built on this
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View a complete list of semi-basic definitions on this wiki
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 defines a subgroup property that makes sense within a finite group
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
ECD condition
The property of being a -Sylow subgroup is obtained as the property of being maximal corresponding to the group property of being a -Sylow subgroup. It turns out that:
- Existence (E): For every , there exist -Sylow subgroups. For full proof, refer: Sylow subgroups exist
- Domination(D): Any -group is contained in a -Sylow subgroup. For full proof, refer: Sylow implies order-dominating
- Conjugacy(C): Any two -Sylow subgroups are conjugate. For full proof, refer: Sylow implies order-conjugate
All these facts, together, show that the group property of being a -group satisfies the ECD condition.
Relation with other properties
Conjunction with other properties
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
Metaproperties
Metaproperty | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
intermediate subgroup condition | Yes | Sylow satisfies intermediate subgroup condition | If is a Sylow subgroup of , and is any intermediate subgroup of containing , then is a Sylow subgroup of . |
transfer condition | No | Sylow does not satisfy transfer condition | If , with a Sylow subgroup of , need not be a Sylow subgroup of . |
permuting transfer condition | Yes | If with a Sylow subgroup of , and , then is a Sylow subgroup of . | |
image condition | Yes | If is a Sylow subgroup of and is surjective, then is a Sylow subgroup of | 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)