When 
is a prime
number, then a 
-group
is a group, all of whose elements have order some power
of 
. For a finite
group, the equivalent definition is that the number of elements in 
is a power of 
. In fact, every finite group
has subgroups which are 
-groups
by the Sylow theorems, in which case they are called
Sylow p-subgroups.
Sylow proved that every group of this form has a power-commutator representation on 
generators defined by
 |
(1)
|
for 
, 
and
![[a_j,a_i]=product_(k=j+1)^na_k^(beta(i,j,k))](/images/equations/p-Group/NumberedEquation2.svg) |
(2)
|
for 
, 
. If 
is a prime power and 
is the number of groups
of order 
,
then
 |
(3)
|
where
 |
(4)
|
(Higman 1960ab).
-Groups. I. Inequalities." Proc. London Math. Soc. 10,
24-30, 1960a.Higman, G. "Enumerating 
-Groups. II. Problems whose solution is PORC." Proc.
London Math. Soc. 10, 566-582, 1960b.