Sylow theorems
Let $G$ be a finite group^{} whose order is divisible by the prime $p$. Suppose ${p}^{m}$ is the highest power of $p$ which is a factor of $G$ and set
$$k=\frac{G}{{p}^{m}}.$$ 
Then

1.
the group $G$ contains at least one subgroup^{} of order ${p}^{m}$,

2.
any two subgroups of $G$ of order ${p}^{m}$ are conjugate^{}, and

3.
the number of subgroups of $G$ of order ${p}^{m}$ is congruent^{} to $1$ modulo $p$ and is a factor of $k$.
Title  Sylow theorems^{} 

Canonical name  SylowTheorems 
Date of creation  20130322 12:24:12 
Last modified on  20130322 12:24:12 
Owner  yark (2760) 
Last modified by  yark (2760) 
Numerical id  6 
Author  yark (2760) 
Entry type  Theorem 
Classification  msc 20D20 
Related topic  SylowPSubgroup 
Related topic  ApplicationOfSylowsTheoremsToGroupsOfOrderPq 
Related topic  SylowsFirstTheorem 
Related topic  SylowsThirdTheorem 
Related topic  SylowPSubgroups 