Math 2.0

Group theory and generalizations Classification of finite simple groups Object

Fischer groups

Builds on Sporadic groupSimple groupOrder of a groupConjugacy classCommutator subgroup

Description

The Fischer groups Fi22\mathrm{Fi}_{22}, Fi23\mathrm{Fi}_{23} and Fi24′\mathrm{Fi}_{24}' are three sporadic groups, found by Bernd Fischer. Their orders are:

  1. ∣Fi22∣=64561751654400=217⋅39⋅52⋅7⋅11⋅13\abs{\mathrm{Fi}_{22}} = 64561751654400 = 2^{17} \cdot 3^9 \cdot 5^2 \cdot 7 \cdot 11 \cdot 13;
  2. ∣Fi23∣=4089470473293004800=218⋅313⋅52⋅7⋅11⋅13⋅17⋅23\abs{\mathrm{Fi}_{23}} = 4089470473293004800 = 2^{18} \cdot 3^{13} \cdot 5^2 \cdot 7 \cdot 11 \cdot 13 \cdot 17 \cdot 23;
  3. ∣Fi24′∣=1255205709190661721292800=221⋅316⋅52⋅73⋅11⋅13⋅17⋅23⋅29\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.

Remarks

Fischer found these groups while classifying 3-transposition groups. A 3-transposition group is a group generated by a conjugacy class DD of involutions such that the product of any two elements of DD has order 11, 22 or 33. The symmetric groups, with DD the class of transpositions, are the basic examples. The group Fi24\mathrm{Fi}_{24} generated by Fischer's class of 3-transpositions is not simple; its commutator subgroup Fi24′\mathrm{Fi}_{24}' has index 22 and is simple. All three Fischer groups are subquotients of the Monster group.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

Show the path as a graph
Commutator subgroup, concept, no evidence levelCommutator subgroupconceptConjugacy class, concept, no evidence levelConjugacy classconceptOrder of a group, concept, no evidence levelOrder of a groupconceptSimple group, concept, no evidence levelSimple groupconceptSporadic group, concept, no evidence levelSporadic groupconceptFischer groups, object, no evidence levelFischer groupsobject
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Commutator subgroup concept
  2. Conjugacy class concept
  3. Order of a group concept
  4. Simple group concept
  5. Sporadic group concept
  6. Fischer groups object

Further reading

  1. Web https://en.wikipedia.org/wiki/Fischer_group
About this page
TagAYJ1CM1XT8
URIm2://AYJ1CM1XT8
Kindobject
WritingW0 stub
MSC20D08
UsesSporadic group, Simple group, Order of a group, Conjugacy class, Commutator subgroup
Same as
Changed2 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.