# Janko group

In mathematics, a Janko group is one of the four sporadic simple groups named for 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.

## Janko groups

• The Janko group J1 has order 175 560 = 23 ···· 11 · 19. It is the only Janko group whose existence was proved by Janko himself.
• The Hall–Janko group has order 604 800 = 27 · 33 · 52 · 7. It is also known as J2, HJ, or the Hall–Janko–Wales group. It was constructed by Marshall Hall, Jr. and David Wales.
• The Janko group J3 has order 50 232 960 = 27 · 35 ·· 17 · 19. It is also known as the Higman–Janko–McKay group. It was constructed by Graham Higman and John McKay.
• The Janko group J4 has order 86 775 571 046 077 562 880 = 221 · 33 ··· 113 · 23 · 29 · 31 · 37 · 43. It was constructed by Simon P. Norton.

## References

1. ^ Dieter Held, Die Klassifikation der endlichen einfachen Gruppen (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." [1]