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Given a positive integer
, it is not a simple matter to determine how many isomorphism types of groups of order
there are.
If
is the
square of a prime, then there are exactly two possible isomorphism types of groups of order
, both of which are Abelian. If
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
, 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
, some restrictions may be placed on the structure of groups of order
, as a consequence, for example, of results such as the
Sylow theorems. For example, every group of order
is cyclic when
are primes with
not divisible by
. The
cyclic numbers (
such that there is just one group of order
) are the numbers
such that
and
are
relatively prime.
Solvable groups
If
is
squarefree, then any group of order
is solvable. A theorem of William Burnside, proved using group characters, states that every group of order
is solvable when
is divisible by fewer than three distinct primes. By the
Feit–Thompson theorem, which has a long and complicated proof, every group of order
is solvable when
is odd.
For every positive integer
, most groups of order
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
solvable groups of order
) but the proof of this for all orders uses the
classification of finite simple groups.
Simple groups
For any integer there are at most
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
is
multiplicative, i.e.
a (m n) = a (m) a (n), (m, n) = 1 |
. The number of Abelian groups of order
(
prime powers), is the
number of partitions of
(
A000041). Thus
where
is the number of partitions of
.
Number of non-Abelian groups of order n
(...)
Table of number of distinct groups of order n
Number of distinct groups of order
Order
|
Prime factorization of
|
|
Number of groups[1]
|
Number of simple groups
|
Number of Abelian groups
|
Number of non-Abelian groups
|
Number of solvable groups
|
Number of non-solvable groups
|
Comment
|
1
|
|
0
|
1
|
|
1
|
0
|
|
0
|
|
2
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
3
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
4
|
|
1
|
2
|
|
2
|
0
|
|
|
Order
|
5
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
6
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
7
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
8
|
|
1
|
5
|
|
3
|
2
|
|
|
Order
|
9
|
|
1
|
2
|
|
2
|
0
|
|
0
|
Order
|
10
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
11
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
12
|
|
2
|
5
|
|
2
|
3
|
|
|
NOT squarefree
|
13
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
14
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
15
|
|
2
|
1
|
|
1
|
0
|
|
0
|
Order with primes and NOT divisible by
|
16
|
|
1
|
14
|
|
5
|
9
|
|
|
Order
|
17
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
18
|
|
2
|
5
|
|
2
|
3
|
|
|
NOT squarefree
|
19
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
20
|
|
2
|
5
|
|
2
|
3
|
|
|
NOT squarefree
|
21
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
22
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
23
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
24
|
|
2
|
15
|
|
3
|
12
|
|
|
NOT squarefree
|
25
|
|
1
|
2
|
|
2
|
0
|
|
0
|
Order
|
26
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
27
|
|
1
|
5
|
|
3
|
2
|
|
0
|
Order
|
28
|
|
2
|
4
|
|
2
|
2
|
|
|
NOT squarefree
|
29
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
30
|
|
3
|
4
|
|
1
|
3
|
|
|
Order
|
31
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
32
|
|
1
|
51
|
|
7
|
44
|
|
|
Order
|
33
|
|
2
|
1
|
|
1
|
0
|
|
0
|
Order with primes and NOT divisible by
|
34
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
35
|
|
2
|
1
|
|
1
|
0
|
|
0
|
Order with primes and NOT divisible by
|
36
|
|
2
|
14
|
|
4
|
10
|
|
|
NOT squarefree
|
37
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
38
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
39
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
40
|
|
2
|
14
|
|
3
|
11
|
|
|
NOT squarefree
|
41
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
42
|
|
3
|
6
|
|
1
|
5
|
|
|
Order
|
43
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
44
|
|
2
|
4
|
|
2
|
2
|
|
|
NOT squarefree
|
45
|
|
2
|
2
|
|
2
|
0
|
|
0
|
NOT squarefree
|
46
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
47
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
48
|
|
2
|
52
|
|
5
|
47
|
|
|
NOT squarefree
|
49
|
|
1
|
2
|
|
2
|
0
|
|
0
|
Order
|
50
|
|
2
|
5
|
|
2
|
3
|
|
|
NOT squarefree
|
51
|
|
2
|
1
|
|
1
|
0
|
|
0
|
Order with primes and NOT divisible by
|
52
|
|
2
|
5
|
|
2
|
3
|
|
|
NOT squarefree
|
53
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
54
|
|
2
|
15
|
|
3
|
12
|
|
|
NOT squarefree
|
55
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
56
|
|
2
|
13
|
|
3
|
10
|
|
|
NOT squarefree
|
57
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
58
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
59
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
60
|
|
3
|
13
|
|
2
|
11
|
12
|
1
|
NOT squarefree
|
61
|
|
1
|
1
|
|
1
|
0
|
|
0
|
Order
|
62
|
|
2
|
2
|
|
1
|
1
|
|
0
|
Order with primes and divisible by
|
63
|
|
2
|
4
|
|
2
|
2
|
|
0
|
NOT squarefree
|
64
|
|
1
|
267
|
|
11
|
256
|
|
|
Order
|
Sequences
A000001 Number of groups of order
.
-
{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.
such that
and
are
relatively prime; also
such that there is just one group of order
, i.e.
A000001 .
-
{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
.
-
{1, 1, 2, 5, 14, 51, 267, 2328, 56092, 10494213, 49487365422, ...} |
A000688 Number of Abelian groups of order
. (Number of factorizations of
into
prime powers greater than
.)
-
{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
.
-
{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
minus the number of non-Abelian groups of order
.
-
{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):
such that every group of order
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):
such that some group of order
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
.
-
A000113 Number of
transformation groups of order
.
-
A000041 Number of partitions of
(the partition numbers).
-
See also
Notes
- ↑ Humphreys, John F. (1996). A Course in Group Theory. Oxford University Press. pp. 238-242.