A 
-group, where 
is a prime number, 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 
. The Sylow
theorems guarantee subgroups whose order
is the largest power of 
dividing the order of the finite group. These are called Sylow
p-subgroups.
A finite group of order 
has a power-commutator
group presentation on 
generators, in which
 |
(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 1960a, Sims 1965).
For each fixed prime number 
, Su and Xia (2026) reported that almost all isomorphism
classes of groups of group order 
have nilpotency class 2 as 
. This means that every commutator
is central and the group is not Abelian.
For a nonnegative integer 
, Pyber asked whether, if every Abelian normal subgroup of a finite 
-group 
has order at most 
, every Abelian subgroup
must have order at most 
(Muliarchyk 2026, p. 1). Muliarchyk (2026, p. 5)
gives a counterexample of group
order 
in which every Abelian normal
subgroup has order at most 
, while an elementary
Abelian group occurs as a subgroup of order 
. The same group
refutes the corresponding bound of 
on the number of generators
needed for Abelian subgroups
when every Abelian normal
subgroup needs at most 
generators. The group
of group order 
is obtained by a counting argument, without a minimality
claim.
-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.Muliarchyk, K. "Abelian
Subgroups of Finite 2-Groups: Counterexamples to Problems 20.75 and 20.76."
2026. 
-Groups."
Proc. London Math. Soc. 15, 151-166, 1965. 
-Groups of Class Two." 8 Oct 2026.