##  [Sylow Theorems](/sylow-theorems) 

  ##  [Sylow Theorems](https://mathlogic.quantumdictionary.io/sylow-theorems-0) 

  

 [![Mathematics & Logic Dictionary](/sites/default/files/styles/large/public/2026-01/Mathematics%20%26%20Logic.png.webp?itok=UhtTRPnp)](/topic-specific-dictionaries/natural-formal-sciences/mathematics-logic)

- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A collection of three fundamental results about p-subgroups of a finite group: existence (there is a subgroup of order the maximal power of a prime p dividing the group order), conjugacy (all such maximal p-subgroups are conjugate), and counting (the number of these subgroups satisfies congruence and divisibility constraints).

 

 

 

 

 





 

 



 ##  [Sylow Theorems](https://puremath.quantumdictionary.io/sylow-theorems-1) 

  

 [![Pure Mathematics Dictionary](/sites/default/files/styles/large/public/2026-01/Pure%20Mathematics.png.webp?itok=5pZnFQ59)](/topic-specific-dictionaries/mathematics-logic/pure-mathematics)

- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A collection of theorems describing, for a finite group and a prime p dividing its order, the existence of p-subgroups of maximal p-power order (Sylow p-subgroups), their conjugacy properties, and congruence/counting constraints on their number.

 

 

 

 

 





 

 



 ##  [Sylow Theorems](https://algebra.quantumdictionary.io/sylow-theorems-2) 

  

 [![Algebra](/sites/default/files/styles/large/public/2026-01/Algebra.png.webp?itok=3pHxBnUF)](/topic-specific-dictionaries/pure-mathematics/algebra)

- Pure Mathematics -

**Algebra Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A trio of fundamental results about p-subgroups in finite groups: existence (for a prime p dividing the group order there exists a subgroup of order p^n where p^n is the maximal p-power dividing |G|), conjugacy (all Sylow p-subgroups are conjugate), and counting (the number n_p of Sylow p-subgroups satisfies n_p ≡ 1 (mod p) and divides the p′-part of |G|).