# Sylow theorems

1. Jan 11, 2009

### SNOOTCHIEBOOCHEE

1. The problem statement, all variables and given/known data

Let G be a group of order pem. Prove that G contains a subgroup of order pr for every integer r $$\leq$$e

3. The attempt at a solution

By sylow 1 we know that there exists a subgroup of order pe. so that case is done, we need to show that supgroups of order r<e exist now.

Thats tricky. im guessing this is an application of the the thrid sylow theorem but i dont know how to use it.

s is the number of sylow p=subgroups. then s|m and is congruent to 1 mod p.

so we know that s has to be at least e. dunno where to go from here.