Pariah group explained

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

size! class="unsortable"
Factorized order	First missing prime in order
Lyons group, Ly		5	2⁸ · 3⁷ · 5⁶ · 7 · 11 · 31 · 37 · 67	13
O'Nan group, O'N		5	2⁹ · 3⁴ · 5 · 7³ · 11 · 19 · 31	13
Rudvalis group, Ru		1	2¹⁴ · 3³ · 5³ · 7 · 13 · 29	11
Janko group, J₄		9	2²¹ · 3³ · 5 · 7 · 11³ · 23 · 29 · 31 · 37 · 43	13
Janko group, J₃		5	2⁷ · 3⁵ · 5 · 17 · 19	7
Janko group, J₁		2	2³ · 3 · 5 · 7 · 11 · 19	13

Lyons group

See main article: article and Lyons group. The Lyons group,

, is the unique group (up to isomorphism) that has in involution

where

C_G(t)

is the covering group of the alternating group

A₁₁

, and

is not weakly closed in

C_G(t)

. Richard Lyons, the namesake of these groups, was the first to consider their properties, including their order, and Charles Sims proved with machine calculation that such a group must exist and be unique. The group has an order of

2⁸ ⋅ 3⁷ ⋅ 5⁶ ⋅ 7 ⋅ 11 ⋅ 31 ⋅ 37 ⋅ 67

.^[1]

Rudvalis group

See main article: article and Rudvalis group. The Rudvalis group is a finite simple 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, who proved the existence of such a group. This group has order of

145,926,144,000=2¹⁴ ⋅ 3³ ⋅ 5³ ⋅ 7 ⋅ 13 ⋅ 29

.^[2]

Janko groups

J₁

References

- Robert A. Wilson (1986). Is J₁ a subgroup of the monster?, Bull. London Math. Soc. 18, no. 4 (1986), 349-350

Notes and References

Aschbacher . Michael . Segev . Yoav . The Uniqueness of Groups of Lyons Type . Journal of the American Mathematical Society . 1992 . 5 . 1 . 75–98 . 10.2307/2152751 . 0894-0347.
Conway . J.H . Wales . D.B . Construction of the Rudvalis group of order 145,926,144,000 . Journal of Algebra . December 1973 . 27 . 3 . 538–548.