|
PRIME FACTORS AND THEIR EXPONENTS IN THE SPORADIC FINITE SIMPLE GROUPS |
|
JUNE 30TH 2010 |
|
|
| |
2 |
3 |
5 |
7 |
11 |
13 |
17 |
19 |
23 |
29 |
31 |
37 |
41 |
43 |
47 |
53 |
59 |
61 |
67 |
71 |
| M11 |
4 |
2 |
1 |
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| M12 |
6 |
3 |
1 |
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| M22 |
7 |
2 |
1 |
1 |
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| M23 |
7 |
2 |
1 |
1 |
1 |
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
| M24 |
10 |
3 |
1 |
1 |
1 |
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
| J1 |
3 |
1 |
1 |
1 |
1 |
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
| J2 = HJ |
7 |
3 |
2 |
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| J3 |
7 |
5 |
1 |
|
|
|
1 |
1 |
|
|
|
|
|
|
|
|
|
|
|
|
| J4 |
21 |
3 |
1 |
1 |
3 |
|
|
|
1 |
1 |
1 |
1 |
|
1 |
|
|
|
|
|
|
| HS |
9 |
2 |
3 |
1 |
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| McL |
7 |
6 |
3 |
1 |
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| He |
10 |
3 |
2 |
3 |
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
| Ru |
14 |
3 |
3 |
1 |
|
1 |
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
| Suz |
13 |
7 |
2 |
1 |
1 |
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| O'N |
9 |
4 |
1 |
3 |
1 |
|
|
1 |
|
|
1 |
|
|
|
|
|
|
|
|
|
| Co3 = .3 |
10 |
7 |
3 |
1 |
1 |
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
| Co2 = .2 |
18 |
6 |
3 |
1 |
1 |
|
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
| Co1 = .1 |
21 |
9 |
4 |
2 |
1 |
1 |
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
| F5 = HN |
14 |
6 |
6 |
1 |
1 |
|
|
1 |
|
|
|
|
|
|
|
|
|
|
|
|
| Ly |
8 |
7 |
6 |
1 |
1 |
|
|
|
|
|
1 |
1 |
|
|
|
|
|
|
1 |
|
| F3 = Th |
15 |
10 |
3 |
2 |
|
1 |
|
1 |
|
|
1 |
|
|
|
|
|
|
|
|
|
| Fi22 |
17 |
9 |
2 |
1 |
1 |
1 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
| Fi23 |
18 |
13 |
2 |
1 |
1 |
1 |
1 |
|
1 |
|
|
|
|
|
|
|
|
|
|
|
| Fi24' |
21 |
16 |
2 |
3 |
1 |
1 |
1 |
|
1 |
1 |
|
|
|
|
|
|
|
|
|
|
| F2 = B |
41 |
13 |
6 |
2 |
1 |
1 |
1 |
1 |
1 |
|
1 |
|
|
|
1 |
|
|
|
|
|
| F1 = M |
46 |
20 |
9 |
6 |
2 |
3 |
1 |
1 |
1 |
1 |
1 |
|
1 |
|
1 |
|
1 |
|
|
1 |
| |
2 |
3 |
5 |
7 |
11 |
13 |
17 |
19 |
23 |
29 |
31 |
37 |
41 |
43 |
47 |
53 |
59 |
61 |
67 |
71 |
|
|
General Wikipedia lemmas on sporadic groups:
http://en.wikipedia.org/wiki/Sporadic_group and http://en.wikipedia.org/wiki/Category:Sporadic_groups
The pariahs among the sporadic finite simple groups are the six groups not involved in the Monster group.
The pariah prime numbers are those which do not appear as prime factors in the orders of the sporadic finite simple groups. They are 53, 61 and all primes from 73 onwards. The ratio of pariah primes against the happy family of primes is infinity against eighteen, so one creates an oxymoron by this definition.
The GCD of the orders of all sporadic finite simple groups is 120.
Finite simple groups in Wikipedia: list - classification
|
|
|
|