Number of groups of given order

Definition

Let $n$ be a natural number. The number of groups of order $n$ is defined as the number of isomorphism classes of groups whose order is $n$ .

This is a finite number and is bounded by $n^{2 n}$ for obvious reasons. The function is not strictly increasing in $n$ and depends heavily on the nature of the prime factorization of $n$ .

Facts

Basic facts

Value of $n$	What we can say about the number of groups of order $n$	Explanation
1	1	only the trivial group
$p$ a prime number	1	only the group of prime order. See equivalence of definitions of group of prime order
$p^{2}$ , $p$ prime	2	only the cyclic group of prime-square order and the elementary abelian group of prime-square order
$p^{3}$ , $p$ prime	5	see classification of groups of prime-cube order
$2^{4} = 16$	14	see classification of groups of order 16, also groups of order 16 for summary information.
$p^{4}$ , $p$ odd prime	15	see classification of groups of prime-fourth order for odd prime
product $p_{1} p_{2} \dots p_{n}$ , $p_{i}$ distinct primes with no $p_{i}$ dividing $p_{j} - 1$	1	the cyclic group of that order. See classification of cyclicity-forcing numbers

Asymptotic fact

Higman-Sims asymptotic formula on number of groups of prime power order
Pyber's theorem on logarithmic quotient of number of nilpotent groups to number of groups approaching unity

Properties

Supermultiplicativity

If $n = a b$ with $a$ and $b$ relatively prime, the number of groups of order $n$ is bounded from below by the product of the number of groups of orders $a$ and $b$ respectively. This is because we can take direct products for every pair of a group of order $a$ and a group of order $b$ .