---
tag: 390Z8QV5ZG
kind: definition
name: Sporadic group
msc: [20D08, 20D05]
defines: PZ49Z4CBFZ
uses: [0YMYMNQ0QZ, 0KVGH0J5B3, 0Z5905Q73D, NPEBABQ49M]
sources:
  - url: https://en.wikipedia.org/wiki/Sporadic_group
---

## Definition

Let $G$ be a finite [[simple group|0YMYMNQ0QZ]].
The group $G$ is *sporadic* when it is not isomorphic to any of the following:

1. a [[cyclic group|0KVGH0J5B3]] of prime order;
2. an [[alternating group|0Z5905Q73D]] $A_n$ with $n \ge 5$;
3. a simple [[group of Lie type|NPEBABQ49M]].

## Remarks

By the [[classification of finite simple groups|06J6P03MG7]], there are exactly $26$ sporadic groups up to isomorphism.
The [[Tits group|WC16TBY3FN]] ${}^2F_4(2)'$ is usually counted with the groups of Lie type in clause 3; some authors count it as a $27$th sporadic group.
Example: the [[Mathieu group|T1KR4KPX8P]] $M_{11}$ is sporadic.
It is simple and not cyclic, its order $7920$ is not of the form $n!/2$, and no simple group of Lie type has order $7920$.
Non-example: $A_5$ is simple but not sporadic, since it is an alternating group, so clause 2 fails.
The sporadic groups include the five Mathieu groups, the four [[Janko groups|PSEC8D7CX0]], the three [[Conway groups|366D4ETGQH]] and the three [[Fischer groups|AYJ1CM1XT8]].
The others are the Higman–Sims, McLaughlin, Held, Rudvalis, Suzuki, O'Nan, Harada–Norton, Lyons and Thompson groups, the [[Baby Monster|TNPQ9AWVBC]] and the [[Monster|C9VY20TAPT]].

## Explanations

### L1 human

A sporadic group is a finite simple group that does not belong to the regular infinite families.