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))](https://mathworld.wolfram.com/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.