# 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

## Contents

## Definition

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

## Behavior

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

### Initial values

For the number of groups is .

### Lower bound

The numbers for which there is only one isomorphism class of groups of order 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 divides for prime divisors . In particular, all prime numbers are cyclicity-forcing, and thus, the number of groups function takes the value for infinitely many natural numbers. Note that there *are* composite cyclicity-forcing numbers, the smallest of which is .

### 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.