Number of groups of given order

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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^{n^2} for obvious reasons. The function is not strictly increasing in n and depends heavily on the nature of the prime factorization of n.

Initial values

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

Numbers up till 100

n Number of groups of order n Reason/explanation
1 1
2 1 prime number
3 1 prime number
4 2 square of a prime; see classification of groups of prime-square order
5 1 prime number
6 2 form pq where p,q primes, q \mid p - 1
7 1 prime number
8 5 prime cube: classification of groups of prime-cube order, also see groups of order 8
9 2 prime square; see classification of groups of prime-square order
Orders 10 to 36. We omit the prime numbers since there is only one group of each such order.[SHOW MORE] Orders greater than 36. We omit prime numbers, squares of primes, and numbers of the form pq where p,q both primes, since these are covered by standard cases.[SHOW MORE]

Small powers of small primes

(For general formulas, see the next section).

Powers of 2:[SHOW MORE]


Powers of 3: [SHOW MORE]


Powers of 5: [SHOW MORE]


Powers of 7: [SHOW MORE]

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
2^5 = 32 51
3^5 = 243 67
p^5, prime p \ge 5 2p + 61 + 2\operatorname{gcd}(p-1,3) + \operatorname{gcd}(p-1,4)
p^6, prime p \ge 5 3p^2 + 39p + 344 + 24 \operatorname{gcd}(p - 1,3) + 11 \operatorname{gcd}(p-1,4) + 2 \operatorname{gcd}(p-1,5)
product p_1p_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
product pq, p,q primes with p dividing q - 1 2
product 4p, p prime, p > 3, p \equiv -1 \pmod 4 4
product 4p, p prime, p \equiv 1 \pmod 4 5

Asymptotic facts and conjectures

Properties

Supermultiplicativity

If n = ab 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.