# 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.

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.