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 (white ellipses) 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. Three other sporadic groups were also shown to be pariahs by Griess in 1982, and the Janko Group J₁ was shown to be the final pariah by Robert A. Wilson in 1986. The complete list is shown below.

List of pariah groups

Group	Size	Approx. size	Factorized order	First missing prime in order
Lyons group, Ly	51765179004000000	5×1016	28 · 37 · 56 · 7 · 11 · 31 · 37 · 67	13
O'Nan group, O'N	460815505920	5×1011	29 · 34 · 5 · 73 · 11 · 19 · 31	13
Rudvalis group, Ru	145926144000	1×1011	214 · 33 · 53 · 7 · 13 · 29	11
Janko group, J4	86775571046077562880	9×1019	221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43	13
Janko group, J3	50232960	5×107	27 · 35 · 5 · 17 · 19	7
Janko group, J1	175560	2×105	23 · 3 · 5 · 7 · 11 · 19	13

References

• Griess, Robert L. (February 1982), "The friendly giant" (PDF), Inventiones Mathematicae, 69 (1): 1–102, Bibcode:1982InMat..69....1G, doi:10.1007/BF01389186, hdl:2027.42/46608, ISSN 0020-9910, MR 0671653, S2CID 123597150
• Robert A. Wilson (1986). Is J₁ a subgroup of the monster?, Bull. London Math. Soc. 18, no. 4 (1986), 349-350