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''Is J1 a subgroup of the monster?''
Bull. London Math. Soc. 18, no. 4 (1986), 349-350 Sporadic groups {{Abstract-algebra-stub
In 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
Robert Louis Griess, Jr. (born 1945, Savannah, Georgia) is a mathematician working on finite simple groups and vertex algebras. He is currently the John Griggs Thompson Distinguished University Professor of mathematics at University of Michigan.
...
in to refer to the six sporadic simple groups
In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups.
A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...
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, having order
246320597611213317192329314147 ...
.
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 {{Abstract-algebra-stub