Pariah group

 ( Sporadic Groups ) 

In group theory, the term pariah was introduced by Robert Griess in to refer to the six sporadic simple groups which are not 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.

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The pariah groups

+ List of pariah groups ! Group ! Size ! class="unsortable"	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

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Lyons group The Lyons group, $Ly$, is the unique group (up to isomorphism) that has in involution $t$ where $C\_G(t)$ is the covering group of the alternating group $A\_\{11\}$, and $t$ 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 $2^8\; \backslash cdot\; 3^7\; \backslash cdot\; 5^6\; \backslash cdot\; 7\; \backslash cdot\; 11\; \backslash cdot\; 31\; \backslash cdot\; 37\; \backslash cdot\; 67$.

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O'Nan group

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

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Janko groups

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J₄

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J₃

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J₁

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

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