Janko group

In the area of modern algebra known as group theory, the Janko groups are the four sporadic simple groups J₁, J₂, J₃ and J₄ introduced by Zvonimir Janko. Unlike the Mathieu groups, Conway groups, or Fischer groups, the Janko groups do not form a series, and the relation among the four groups is mainly historical rather than mathematical.

History

Janko constructed the first of these groups, J₁, in 1965 and predicted the existence of J₂ and J₃. In 1976, he suggested the existence of J₄. Later, J₂, J₃ and J₄ were all shown to exist.

J₁ was the first sporadic simple group discovered in nearly a century: until then only the Mathieu groups were known, M₁₁ and M₁₂ having been found in 1861, and M₂₂, M₂₃ and M₂₄ in 1873. The discovery of J₁ caused a great "sensation"[1] and "surprise"[2] among group theory specialists. This began the modern theory of sporadic groups.

And in a sense, J₄ ended it. It would be the last sporadic group (and, since the non-sporadic families had already been found, the last finite simple group) predicted and discovered, though this could only be said in hindsight when the Classification theorem was completed.

References

Dieter Held, Die Klassifikation der endlichen einfachen Gruppen Archived 2013-06-26 at the Wayback Machine (the classification of the finite simple groups), Forschungsmagazin der Johannes Gutenberg-Universität Mainz 1/86
The group theorist Bertram Huppert said of J₁: "There were a very few things that surprised me in my life... There were only the following two events that really surprised me: the discovery of the first Janko group and the fall of the Berlin Wall."

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[1] Dieter Held, Die Klassifikation der endlichen einfachen Gruppen Archived 2013-06-26 at the Wayback Machine (the classification of the finite simple groups), Forschungsmagazin der Johannes Gutenberg-Universität Mainz 1/86

[2] The group theorist Bertram Huppert said of J₁: "There were a very few things that surprised me in my life... There were only the following two events that really surprised me: the discovery of the first Janko group and the fall of the Berlin Wall."