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

for , and

for , . If is a prime power and is the number of groups of order , then

where

(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.

See also

Group, Group Direct Product, Group Order, Sylow p-Subgroup, Sylow Theorems

Portions of this entry contributed by Todd Rowland

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References

Higman, G. "Enumerating -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. https://kourovkanotebookorg.wordpress.com/wp-content/uploads/2026/09/kourovka_20_75-20_76_solution.pdf.Sims, C. C. "Enumerating -Groups." Proc. London Math. Soc. 15, 151-166, 1965. https://doi.org/10.1112/plms/s3-15.1.151.Su, N. and Xia, B. "Almost All Finite Groups Are -Groups of Class Two." 8 Oct 2026. https://arxiv.org/abs/2610.11092.

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p-Group

Cite this as:

Weisstein, Eric W., with contributions by Todd Rowland. "p-Group." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/p-Group.html

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