Janko group

A Janko group in group theory one of the four sporadic groups, which were named after Zvonimir Janko. 1965 was the first Janko Janko group J1, saying at the same time the existence of the Janko group J2 and J3 advance. In 1976, he then suspected the existence of the Janko group J4. The J2, J3, and J4 were later proved by other mathematicians. While the Janko group J2 belongs to the so-called happy family, include the groups J1, J3 and J4 to the pariahs. This means that these three groups can not be represented as subgroups or quotient groups of subgroups of the Monster group.

The four Janko groups

• The Janko group J1 has order 175 560 = 23 · 3 · 5 · 7 · 11 · 19 It is the only Janko group whose existence was proved by Janko itself.
• The Janko group J2 is of order 604 800 = 27 · 33 · 52 · 7 It was designed by Marshall Hall and David Wales.
• The Janko group J3 is of order 50,232,960 = 27 · 35 · 5 · 17 · 19 and was designed by Graham Higman and John McKay.
• The Janko group J4 has order 86 775 571 046 077 562 880 constructed = 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 and was from Simon Norton.