Number of groups of order n

This article needs more work.

Please help by expanding it!

Given a positive integer 

n

, it is not a simple matter to determine how many isomorphism types of groups of order 

n

there are. If 

n

is the square of a prime, then there are exactly two possible isomorphism types of groups of order 

n

, both of which are Abelian. If 

n

is a higher power of a prime, then results of Graham Higman and Charles Sims give asymptotically correct estimates for the number of isomorphism types of groups of order 

n

, and the number grows very rapidly as the power increases.

Cyclic groups

Every group of prime order is cyclic, since Lagrange's theorem implies that the cyclic subgroup generated by any of its non-identity elements is the whole group. Depending on the prime factorization of 

n

, some restrictions may be placed on the structure of groups of order 

n

, as a consequence, for example, of results such as the Sylow theorems. For example, every group of order 

p q

is cyclic when 

p < q

are primes with 

q  −  1

not divisible by 

p

. The cyclic numbers ( 

n

such that there is just one group of order 

n

) are the numbers 

n

such that 

n

and 

φ (n)

are relatively prime.

Solvable groups

If 

n

is squarefree, then any group of order 

n

is solvable. A theorem of William Burnside, proved using group characters, states that every group of order 

n

is solvable when 

n

is divisible by fewer than three distinct primes. By the Feit–Thompson theorem, which has a long and complicated proof, every group of order 

n

is solvable when 

n

is odd. For every positive integer 

n

, most groups of order 

n

are solvable. To see this for any particular order is usually not difficult (for example, there is, up to isomorphism, one non-solvable group and 

12

solvable groups of order 

60

) but the proof of this for all orders uses the classification of finite simple groups.

Simple groups

For any integer there are at most 

2

simple groups of that order, and there are infinitely many pairs of non-isomorphic simple groups of the same order.

Number of Abelian groups of order n

The number of Abelian groups of order 

n

is multiplicative, i.e. 

a (m n) = a (m) a (n), (m, n) = 1

. The number of Abelian groups of order 

p k

(prime powers), is the number of partitions of 

k

(A000041). Thus

$a(n)=a{\bigg (}\prod _{i=1}^{\omega (n)}{p_{i}}^{\alpha _{i}}{\bigg )}=\prod _{i=1}^{\omega (n)}p(\alpha _{i}),\,$

where 

p (k)

is the number of partitions of 

k

.

Number of non-Abelian groups of order n

(...)

Table of number of distinct groups of order n

Number of distinct groups of order 

n

Order   n	Prime factorization of  n	ω (n)	Number of groups	Number of simple groups	Number of Abelian groups   ∏ ω (n) i  = 1  p (αi)	Number of non-Abelian groups	Number of solvable groups	Number of non-solvable groups	Comment
1		0	1		1	0		0
2	2 1	1	1		1	0		0	Order  p
3	3 1	1	1		1	0		0	Order  p
4	2 2	1	2		2	0			Order  p 2
5	5 1	1	1		1	0		0	Order  p
6	2 1  ⋅   3 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
7	7 1	1	1		1	0		0	Order  p
8	2 3	1	5		3	2			Order  p 3
9	3 2	1	2		2	0		0	Order  p 2
10	2 1  ⋅   5 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
11	11 1	1	1		1	0		0	Order  p
12	2 2  ⋅   3 1	2	5		2	3			NOT squarefree
13	13 1	1	1		1	0		0	Order  p
14	2 1  ⋅   7 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
15	3 1  ⋅   5 1	2	1		1	0		0	Order  p q with  p < q primes and  q  −  1 NOT divisible by  p
16	2 4	1	14		5	9			Order  p 4
17	17 1	1	1		1	0		0	Order  p
18	2 1  ⋅   3 2	2	5		2	3			NOT squarefree
19	19 1	1	1		1	0		0	Order  p
20	2 2  ⋅   5 1	2	5		2	3			NOT squarefree
21	3 1  ⋅   7 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
22	2 1  ⋅   11 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
23	23 1	1	1		1	0		0	Order  p
24	2 3  ⋅   3 1	2	15		3	12			NOT squarefree
25	5 2	1	2		2	0		0	Order  p 2
26	2 1  ⋅   13 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
27	3 3	1	5		3	2		0	Order  p 3
28	2 2  ⋅   7 1	2	4		2	2			NOT squarefree
29	29 1	1	1		1	0		0	Order  p
30	2 1  ⋅   3 1  ⋅   5 1	3	4		1	3			Order  p q r
31	31 1	1	1		1	0		0	Order  p
32	2 5	1	51		7	44			Order  p 5
33	3 1  ⋅   11 1	2	1		1	0		0	Order  p q with  p < q primes and  q  −  1 NOT divisible by  p
34	2 1  ⋅   17 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
35	5 1  ⋅   7 1	2	1		1	0		0	Order  p q with  p < q primes and  q  −  1 NOT divisible by  p
36	2 2  ⋅   3 2	2	14		4	10			NOT squarefree
37	37 1	1	1		1	0		0	Order  p
38	2 1  ⋅   19 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
39	3 1  ⋅   13 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
40	2 3  ⋅   5 1	2	14		3	11			NOT squarefree
41	41 1	1	1		1	0		0	Order  p
42	2 1  ⋅   3 1  ⋅   7 1	3	6		1	5			Order  p q r
43	43 1	1	1		1	0		0	Order  p
44	2 2  ⋅   11 1	2	4		2	2			NOT squarefree
45	3 2  ⋅   5 1	2	2		2	0		0	NOT squarefree
46	2 1  ⋅   23 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
47	47 1	1	1		1	0		0	Order  p
48	2 4  ⋅   3 1	2	52		5	47			NOT squarefree
49	7 2	1	2		2	0		0	Order  p 2
50	2 1  ⋅   5 2	2	5		2	3			NOT squarefree
51	3 1  ⋅   17 1	2	1		1	0		0	Order  p q with  p < q primes and  q  −  1 NOT divisible by  p
52	2 2  ⋅   13 1	2	5		2	3			NOT squarefree
53	53 1	1	1		1	0		0	Order  p
54	2 1  ⋅   3 3	2	15		3	12			NOT squarefree
55	5 1  ⋅   11 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
56	2 3  ⋅   7 1	2	13		3	10			NOT squarefree
57	3 1  ⋅   19 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
58	2 1  ⋅   29 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
59	59 1	1	1		1	0		0	Order  p
60	2 2  ⋅   3 1  ⋅   5 1	3	13		2	11	12	1	NOT squarefree
61	61 1	1	1		1	0		0	Order  p
62	2 1  ⋅   31 1	2	2		1	1		0	Order  p q with  p < q primes and  q  −  1 divisible by  p
63	3 2  ⋅   7 1	2	4		2	2		0	NOT squarefree
64	2 6	1	267		11	256			Order  p 6

Sequences

A000001 Number of groups of order 

n, n   ≥   1

.

 

{1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, ...}

A003277 Cyclic numbers, i.e. 

n

such that 

n

and 

φ (n)

are relatively prime; also 

n

such that there is just one group of order 

n

, i.e. A000001 

(n) = 1

.

 

{1, 2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 33, 35, 37, 41, 43, 47, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 101, 103, 107, 109, 113, 115, 119, 123, 127, 131, 133, 137, 139, ...}

A000679 Number of groups of order 

2 n, n   ≥   0

.

 

{1, 1, 2, 5, 14, 51, 267, 2328, 56092, 10494213, 49487365422, ...}

A000688 Number of Abelian groups of order 

n, n   ≥   1

. (Number of factorizations of 

n

into prime powers greater than 

1

.)

 

{1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 5, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 7, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 5, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 11, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, ...}

A060689 Number of non-Abelian groups of order 

n, n   ≥   1

.

 

{0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 9, 0, 3, 0, 3, 1, 1, 0, 12, 0, 1, 2, 2, 0, 3, 0, 44, 0, 1, 0, 10, 0, 1, 1, 11, 0, 5, 0, 2, 0, 1, 0, 47, 0, 3, 0, 3, 0, 12, 1, 10, 1, 1, 0, 11, 0, 1, 2, 256, 0, 3, 0, 3, 0, 3, 0, 44, ...}

A066295 Number of Abelian groups of order 

n, n   ≥   1,

minus the number of non-Abelian groups of order 

n, n   ≥   1

.

 

{1, 1, 1, 2, 1, 0, 1, 1, 2, 0, 1, −1, 1, 0, 1, −4, 1, −1, 1, −1, 0, 0, 1, −9, 2, 0, 1, 0, 1, −2, 1, −37, 1, 0, 1, −6, 1, 0, 0, −8, 1, −4, 1, 0, 2, 0, 1, −42, 2, −1, 1, −1, 1, −9, 0, −7, 0, 0, 1, −9, 1, 0, 0, −245, 1, −2, 1, −1, 1, −2, ...}

A051532 The Abelian orders (or Abelian numbers): 

n

such that every group of order 

n

is Abelian.

 

{1, 2, 3, 4, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 31, 33, 35, 37, 41, 43, 45, 47, 49, 51, 53, 59, 61, 65, 67, 69, 71, 73, 77, 79, 83, 85, 87, 89, 91, 95, 97, 99, 101, 103, 107, 109, 113, 115, 119, 121, 123, ...}

A060652 The non-Abelian orders (or non-Abelian numbers): 

n

such that some group of order 

n

is non-Abelian.

 

{6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 64, 66, 68, 70, 72, 74, 75, 76, 78, 80, 81, 82, 84, 86, 88, 90, 92, 93, ...}

A034382 Number of labeled Abelian groups of order 

n

.

 

{?, ...}

A000113 Number of transformation groups of order 

n

.

 

{?, ...}

A000041 Number of partitions of 

n

(the partition numbers).

 

{?, ...}

See also

• Classification of finite simple groups