Number of groups of given order

This article defines an arithmetic function or number-theoretic function: a function from the natural numbers to a ring (usually, the ring of integers, rational numbers, real numbers, or complex numbers).
View a complete list of arithmetic functions

Definition

Let $n$ be a natural number. The number of groups of order $n$ is the number of isomorphism classes of groups whose order is $n$. In other words, it is the maixmum possible number of pairwise non-isomorphic groups of order $n$.

Behavior

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A000001

Initial values

For $n=1,2,3,4,5,6,7,8,9,10,11,12$ the number of groups is $1,1,1,2,1,2,1,5,2,2,1,5$.

Lower bound

The numbers $n$ for which there is only one isomorphism class of groups of order $n$ are termed cyclicity-forcing numbers. These are also the only orders of groups whose automorphism group has coprime order to the group itself. Further information: Groupprops:Classification of cyclicity-forcing numbers, Groupprops:Coprime automorphism group implies cyclic with order a cyclicity-forcing number

A number is a cyclicity-forcing number if and only if it is a product of distinct primes such that no $p_{i}$ divides $p_{j}-1$ for prime divisors $p_{i},p_{j}$. In particular, all prime numbers are cyclicity-forcing, and thus, the number of groups function takes the value $1$ for infinitely many natural numbers. Note that there are composite cyclicity-forcing numbers, the smallest of which is $15$.

Upper bound

High values are typically attained at prime powers.

Relation with other arithmetic functions

• Number of nilpotent groups of given order: This is a multiplicative function, unlike the total number of groups of given order.
• Number of abelian groups of given order: This is also a multiplicative function, and is in fact the product of the number of unordered integer partitions for the exponents of each of the prime divisors of the order.