group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, the term pariah was introduced by Robert Griess in to refer to the six

sporadic simple groups In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups. A simpl ...

which are not

subquotient In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, thou ...

s of the

monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group; it has order : : = 2463205976112133171923293 ...

. The twenty groups which are subquotients, including the monster group itself, he dubbed the happy family. For example, the orders of ''J''₄ 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 ''J''₄ 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.

References

* * Robert A. Wilson (1986)
''Is J₁ a subgroup of the monster?''
Bull. London Math. Soc. 18, no. 4 (1986), 349-350 Sporadic groups {{group-theory-stub