In group theory, the term pariah was introduced by Robert Griess in 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 pariah groups

Lyons group

The Lyons group, is the unique group (up to isomorphism) that has in involution is the covering group of the alternating group is not weakly closed. Richard Lyons, the namesake of these groups, was the first to consider their properties, including their order, and Charles Sims.

O'Nan group

Rudvalis group

The Rudvalis group that is a rank 3 permutation group on 4060 letters where the stabilizer of a point is the Ree group. The group was described by Arunas Rudvalis.

Janko groups

J4

J3

J1

References

• Robert A. Wilson (1986). , Bull. London Math. Soc. 18, no. 4 (1986), 349-350