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- 2Initial values
- 3Facts
- 4Properties
Definition
Access database file. 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 |
Numbers Groups And Codes Solution Manual 2017
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 where 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 7: [SHOW MORE]
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 |
Numbers Groups And Codes Solution Manual Download
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 .
- 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 . Adobe acrobat 11 free download.
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