---
tag: AYJ1CM1XT8
kind: object
name: Fischer groups
msc: [20D08]
uses: [PZ49Z4CBFZ, 0YMYMNQ0QZ, 0B85S2V2DR, 0DEBXAJ5SA, 0ASGSTXSBW]
sources:
  - url: https://en.wikipedia.org/wiki/Fischer_group
same_as:
  wikidata: Q3117683
  wikipedia: Fischer group
---

## Description

The Fischer groups $\mathrm{Fi}_{22}$, $\mathrm{Fi}_{23}$ and $\mathrm{Fi}_{24}'$ are three [[sporadic groups|PZ49Z4CBFZ]], found by Bernd Fischer.
Their [[orders|0B85S2V2DR]] are:

1. $\abs{\mathrm{Fi}_{22}} = 64561751654400 = 2^{17} \cdot 3^9 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13$;
2. $\abs{\mathrm{Fi}_{23}} = 4089470473293004800 = 2^{18} \cdot 3^{13} \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 23$;
3. $\abs{\mathrm{Fi}_{24}'} = 1255205709190661721292800 = 2^{21} \cdot 3^{16} \cdot 5^2 \cdot 7^3 \cdot 11 \cdot 13 \cdot 17 \cdot 23 \cdot 29$.

Each of them is a finite [[simple group|0YMYMNQ0QZ]].

## Remarks

Fischer found these groups while classifying 3-transposition groups.
A 3-transposition group is a group generated by a [[conjugacy class|0DEBXAJ5SA]] $D$ of involutions such that the product of any two elements of $D$ has order $1$, $2$ or $3$.
The symmetric groups, with $D$ the class of transpositions, are the basic examples.
The group $\mathrm{Fi}_{24}$ generated by Fischer's class of 3-transpositions is not simple; its [[commutator subgroup|0ASGSTXSBW]] $\mathrm{Fi}_{24}'$ has index $2$ and is simple.
All three Fischer groups are subquotients of the [[Monster group|C9VY20TAPT]].

## Explanations

### L1 human

In a symmetric group, two transpositions multiply to an element of order $1$, $2$ or $3$.
Fischer asked which groups are generated by a class of elements of order $2$ with this property.
Besides symmetric groups and some matrix groups, the answer contains three new simple groups, the Fischer groups.