Pariah group

In group theory, the term pariah was introduced by Robert Griess in Griess (1982) to refer to the six sporadic simple groups which are not subquotients of the monster group.

Relationships among the sporadic simple groups. The monster group M is at the top, and the groups which are descended from it are the happy family.
The six which are not connected by an upward path to M (white ellipses) are the pariahs.

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

List of pariah groups

Group	Size	Approx. size	Factorized order	First missing prime in order
Lyons group, Ly	51765179004000000	5×1016	28 · 37 · 56 · 7 · 11 · 31 · 37 · 67	13
O'Nan group, O'N	460815505920	5×1011	29 · 34 · 5 · 73 · 11 · 19 · 31	13
Rudvalis group, Ru	145926144000	1×1011	214 · 33 · 53 · 7 · 13 · 29	11
Janko group, J4	86775571046077562880	9×1019	221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43	13
Janko group, J3	50232960	5×107	27 · 35 · 5 · 17 · 19	7
Janko group, J1	175560	2×105	23 · 3 · 5 · 7 · 11 · 19	13

Lyons group

Main article: Lyons group

The Lyons group, ${\displaystyle Ly}$, is the unique group (up to isomorphism) that has in involution ${\displaystyle t}$ where ${\displaystyle C_{G}(t)}$ is the covering group of the alternating group ${\displaystyle A_{11}}$, and ${\displaystyle t}$ is not weakly closed in ${\displaystyle C_{G}(t)}$. 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 ${\displaystyle 2^{8}\cdot 3^{7}\cdot 5^{6}\cdot 7\cdot 11\cdot 31\cdot 37\cdot 67}$.^[1]

O'Nan group

This section is an excerpt from O'Nan group.[edit]

In the area of abstract algebra known as group theory, the O'Nan group O'N or O'Nan–Sims group is a sporadic simple group of order

   460,815,505,920 = 2⁹ · 3⁴ · 5 · 7³ · 11 · 19 · 31 ≈ 5×10¹¹.

Rudvalis group

Main article: Rudvalis group

The Rudvalis group is a finite simple group ${\displaystyle R}$ 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 ${\displaystyle 145,926,144,000=2^{14}\cdot 3^{3}\cdot 5^{3}\cdot 7\cdot 13\cdot 29}$.^[2]

Janko groups

J₄

This section is an excerpt from Janko group J4.[edit]

In the area of modern algebra known as group theory, the Janko group J₄ is a sporadic simple group of order

   86,775,571,046,077,562,880

= 2²¹ · 3³ · 5 · 7 · 11³ · 23 · 29 · 31 · 37 · 43

≈ 9×10¹⁹.

J₃

This section is an excerpt from Janko group J3.[edit]

In the area of modern algebra known as group theory, the Janko group J₃ or the Higman-Janko-McKay group HJM is a sporadic simple group of order

   50,232,960 = 2⁷ · 3⁵ · 5 · 17 · 19.

J₁

This section is an excerpt from Janko group J1.[edit]

In the area of modern algebra known as group theory, the Janko group J₁ is a sporadic simple group of order

   175,560 = 2³ · 3 · 5 · 7 · 11 · 19

≈ 2×10⁵.