Number of groups of given order

From Number
Jump to: navigation, search
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


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 .


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