# Janko group

Algebraic structure → Group theoryGroup theory |
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In the area of modern algebra known as group theory, the **Janko groups** are the four sporadic simple groups *J _{1}*,

*J*,

_{2}*J*and

_{3}*J*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.

_{4}## 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 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*(the classification of the finite simple groups), Forschungsmagazin der Johannes Gutenberg-Universität 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." [1]

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Original source: https://en.wikipedia.org/wiki/Janko group.
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