Let 
be a prime number, 
a finite group, and 
the order of 
.
1. If 
divides 
,
then 
has a Sylow p-subgroup.
2. In a finite group, all the Sylow p-subgroups are conjugate for some
fixed 
.
3. The number of Sylow p-subgroups for a fixed 
is congruent to 1 (mod 
).
