---
tag: 9W6SSHGH4F
kind: definition
name: Pariah groups
msc: [20D08]
defines: ZYZ5FYG7QC
uses: [PZ49Z4CBFZ, C9VY20TAPT, 0X1JXT47GA, 0KYX2N69J0]
sources:
  - url: https://en.wikipedia.org/wiki/Pariah_group
---

## Definition

Let $\mathbb{M}$ be the [[Monster group|C9VY20TAPT]].
Let $H$ be a [[subgroup|0X1JXT47GA]] of $\mathbb{M}$ and let $N$ be a normal subgroup of $H$.
A group isomorphic to such a [[quotient group|0KYX2N69J0]] $H/N$ is a subquotient of $\mathbb{M}$.
A *pariah group* is a [[sporadic group|PZ49Z4CBFZ]] that is not a subquotient of $\mathbb{M}$.

## Remarks

There are exactly six pariah groups: $J_1$, $J_3$, $J_4$, the O'Nan group $\mathrm{O'N}$, the Rudvalis group $\mathrm{Ru}$ and the Lyons group $\mathrm{Ly}$.
Example: the [[Janko group|PSEC8D7CX0]] $J_1$ is a pariah.
Non-example: the [[Conway group|366D4ETGQH]] $\mathrm{Co}_1$ is not a pariah.
The Monster contains an involution whose centraliser $H$ has a normal subgroup $N$ of order $2^{25}$ with $H/N \cong \mathrm{Co}_1$, so $\mathrm{Co}_1$ is a subquotient of $\mathbb{M}$.

## Explanations

### L1 human

A pariah is a sporadic simple group that cannot be found inside the Monster, even as a quotient of a subgroup.