Math 2.0

Group theory and generalizations Classification of finite simple groups Object

Janko groups

Builds on Sporadic groupSimple groupOrder of a groupPariah groupsMonster group

Description

The Janko groups J1J_1, J2J_2, J3J_3 and J4J_4 are four sporadic groups, found by Zvonimir Janko. Their orders are:

  1. ∣J1∣=175560=23⋅3⋅5⋅7⋅11⋅19\abs{J_1} = 175560 = 2^3 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 19;
  2. ∣J2∣=604800=27⋅33⋅52⋅7\abs{J_2} = 604800 = 2^7 \cdot 3^3 \cdot 5^2 \cdot 7;
  3. ∣J3∣=50232960=27⋅35⋅5⋅17⋅19\abs{J_3} = 50232960 = 2^7 \cdot 3^5 \cdot 5 \cdot 17 \cdot 19;
  4. ∣J4∣=86775571046077562880=221⋅33⋅5⋅7⋅113⋅23⋅29⋅31⋅37⋅43\abs{J_4} = 86775571046077562880 = 2^{21} \cdot 3^3 \cdot 5 \cdot 7 \cdot 11^3 \cdot 23 \cdot 29 \cdot 31 \cdot 37 \cdot 43.

Each of them is a finite simple group.

Remarks

J1J_1 was the first sporadic group found after the Mathieu groups, about a century later. Janko characterised it by the centraliser of an involution, which is C2×A5C_2 \times A_5. J1J_1 is a subgroup of the Chevalley group G2(𝔽11)G_2(\F_{11}). J2J_2 is also called the Hall–Janko group. J1J_1, J3J_3 and J4J_4 are pariah groups, since they are not subquotients of the Monster group, while J2J_2 is a subquotient of the Monster.

Reading path

To understand this page from the ground up, read these 9 pages in order. The foundations come first.

Show the path as a graph
Griess algebra, object, no evidence levelGriess algebraobjectIrreducible representation, concept, no evidence levelIrreducible represent…conceptOrder of a group, concept, no evidence levelOrder of a groupconceptPariah groups, concept, no evidence levelPariah groupsconceptSimple group, concept, no evidence levelSimple groupconceptSporadic group, concept, no evidence levelSporadic groupconceptj-invariant, concept, no evidence levelj-invariantconceptMonster group, object, no evidence levelMonster groupobjectJanko groups, object, no evidence levelJanko 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. Griess algebra object
  2. Irreducible representation concept
  3. Order of a group concept
  4. Pariah groups concept
  5. Simple group concept
  6. Sporadic group concept
  7. j-invariant concept
  8. Monster group object
  9. Janko groups object

Further reading

  1. Web https://en.wikipedia.org/wiki/Janko_group
About this page
TagPSEC8D7CX0
URIm2://PSEC8D7CX0
Kindobject
WritingW0 stub
MSC20D08
UsesSporadic group, Simple group, Order of a group, Pariah groups, Monster group
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.