Pariah group

Relationships among the sporadic simple groups. The monster group M is at the top, and the groups which are descended from it are the happy family. The six which are not connected by an upward path to M (four at lower right, two at upper left and right) are the pariahs.

In group theory, the term pariah was introduced by Robert Griess in Griess (1982) to refer to the six sporadic simple groups which are not subquotients of the monster group.

The twenty groups which are subquotients, including the monster group itself, he dubbed the happy family.

For example, the orders of J₄ and the Lyons Group Ly are divisible by 37. Since 37 does not divide the order of the monster, these cannot be subquotients of it; thus J₄ and Ly are pariahs. Four other sporadic groups were also shown to be pariahs. The complete list is shown below.

List of pariah groups

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

References

• Griess, Robert L. (February 1982), "The friendly giant" (PDF), Inventiones Mathematicae, 69 (1): 1–102, doi:10.1007/BF01389186, ISSN 0020-9910, MR 0671653