Pariah group
Diagram of sporadic simple groups, showing the pariahs as 4 groups on the right, and 2 groups as extensions of Happy Family groups. The diagram is correct, except that it incorrectly omits a line from M11 to O'N.
In the area of modern algebra known as group theory, the term pariah was introduced by Griess (1982) to refer to the six sporadic simple groups that are not subquotients of the monster group.
The prime 37 divides the order of the Lyons Group Ly. Since 37 does not divide the order of the monster, Ly cannot be a subquotient of it; thus Ly is a pariah. For exactly the same reason, J4 is a pariah. Four other sporadic groups were also shown to be pariahs. The complete list is shown below.
| Group | Size | Factorized order |
|---|---|---|
| Lyons group Ly | ≈ 5×1016 | 28 • 37 • 56 • 7 • 11 • 31 • 37 • 67 |
| O'Nan group O'N | ≈ 5×1011 | 29 • 34 • 5 • 73 • 11 • 19 • 31 |
| Rudvalis group Ru | ≈ 1×1011 | 214 • 33 • 53 • 7 • 13 • 29 |
| Janko group J4 | ≈ 9×1019 | 221 • 33 • 5 • 7 • 113 • 23 • 29 • 31 • 37 • 43 |
| Janko group J3 | ≈ 5×107 | 27 • 35 • 5 • 17 • 19 |
| Janko group J1 | ≈ 2×105 | 23 • 3 • 5 • 7 • 11 • 19 |
The happy family
The other 20 sporadic groups, those which are subquotients of the monster group, are referred to as the happy family.
References
- Griess, Robert L. (1982), "The friendly giant", Inventiones Mathematicae 69 (1): 1–102, doi:10.1007/BF01389186, ISSN 0020-9910, MR 671653
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