Janko group J2

In the area of modern algebra known as group theory, the Janko group J₂ or the Hall-Janko group HJ is a sporadic simple group of order

History and properties

J₂ is one of the 26 Sporadic groups and is also called Hall–Janko–Wales group. In 1969 Zvonimir Janko predicted J₂ as one of two new simple groups having 2¹⁺⁴:A₅ as a centralizer of an involution (the other is the Janko group J3). It was constructed by Hall and Wales (1968) as a rank 3 permutation group on 100 points.

Both the Schur multiplier and the outer automorphism group have order 2.

J₂ is the only one of the 4 Janko groups that is a subquotient of the monster group; it is thus part of what Robert Griess calls the Happy Family. Since it is also found in the Conway group Co1, it is therefore part of the second generation of the Happy Family.

Representations

It is a subgroup of index two of the group of automorphisms of the Hall–Janko graph, leading to a permutation representation of degree 100. It is also a subgroup of index two of the group of automorphisms of the Hall–Janko Near Octagon, leading to a permutation representation of degree 315.

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" and "surprise" 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.