# 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. 1.

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

2. 2.

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

3. 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 SylowTheorems 2013-03-22 12:24:12 2013-03-22 12:24:12 yark (2760) yark (2760) 6 yark (2760) Theorem msc 20D20 SylowPSubgroup ApplicationOfSylowsTheoremsToGroupsOfOrderPq SylowsFirstTheorem SylowsThirdTheorem SylowPSubgroups