Janko group
Algebraic structure → Group theory Group theory 


Modular groups

Infinite dimensional Lie group

In the area of modern algebra known as group theory, the Janko groups are the four sporadic simple groups J_{1}, J_{2}, J_{3} and J_{4} 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, J_{1}, in 1965 and predicted the existence of J_{2} and J_{3}. In 1976, he suggested the existence of J_{4}. Later, J_{2}, J_{3} and J_{4} were all shown to exist.
J_{1} was the first sporadic simple group discovered in nearly a century: until then only the Mathieu groups were known, M_{11} and M_{12} having been found in 1861, and M_{22}, M_{23} and M_{24} in 1873. The discovery of J_{1} caused a great "sensation"^{[1]} and "surprise"^{[2]} among group theory specialists. This began the modern theory of sporadic groups.
And in a sense, J_{4} ended it. It would be the last sporadic group (and, since the nonsporadic 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 (the classification of the finite simple groups), Forschungsmagazin der Johannes GutenbergUniversität Mainz 1/86
 ↑ The group theorist Bertram Huppert said of J_{1}: "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."