# Number of groups of given order

From Groupprops

## Contents

## Definition

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

This is a finite number and is bounded by for obvious reasons. The function is *not* strictly increasing in and depends heavily on the nature of the prime factorization of .

## Initial values

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

### Numbers up till 100

Number of groups of order | 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 where primes, |

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 |

### Small powers of small primes

(For general formulas, see the next section).

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## Facts

### Basic facts

Value of | What we can say about the number of groups of order | Explanation |
---|---|---|

1 | 1 | only the trivial group |

a prime number | 1 | only the group of prime order. See equivalence of definitions of group of prime order |

, prime | 2 | only the cyclic group of prime-square order and the elementary abelian group of prime-square order |

, prime | 5 | see classification of groups of prime-cube order |

14 | see classification of groups of order 16, also groups of order 16 for summary information. | |

, odd prime | 15 | see classification of groups of prime-fourth order for odd prime |

51 | ||

67 | ||

, prime | ||

, prime | ||

product , distinct primes with no dividing | 1 | the cyclic group of that order. See classification of cyclicity-forcing numbers |

product , primes with dividing | 2 | |

product , prime, , | 4 | |

product , prime, | 5 |

### Asymptotic facts and conjectures

- Higman-Sims asymptotic formula on number of groups of prime power order: This states that the number of groups of order is about .
- Conjecture that most finite groups are nilpotent
- Pyber's theorem on logarithmic quotient of number of nilpotent groups to number of groups approaching unity
- Higman's PORC conjecture states that the number of groups of order is a PORC function in for fixed .

## Properties

### Supermultiplicativity

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