In the area of modern algebra known as

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

, the Janko groups are the four

sporadic simple group In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The ...

s '' J₁'', '' J₂'', '' J₃'' and '' J₄'' introduced by

Zvonimir Janko Zvonimir Janko (26 July 1932 – 12 April 2022) was a Croatian mathematician who was the eponym of the Janko groups, sporadic simple groups in group theory. The first few sporadic simple groups were discovered by Émile Léonard Mathieu, which ...

. Unlike the

Mathieu groups In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 object ...

Conway groups In the area of modern algebra known as group theory, the Conway groups are the three sporadic simple groups Co1, Co2 and Co3 along with the related finite group Co0 introduced by . The largest of the Conway groups, Co0, is the group of auto ...

, or

Fischer groups In the area of modern algebra known as group theory, the Fischer groups are the three sporadic simple groups Fi22, Fi23 and Fi24 introduced by . 3-transposition groups The Fischer groups are named after Bernd Fischer who discovered them ...

, 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 group In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups ''M''11, ''M''12, ''M''22, ''M''23 and ''M''24 introduced by . They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objec ...

s 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" and "surprise"The group theorist

Bertram Huppert Bertram Huppert (born 22 October 1927 in Worms, Germany) is a German mathematician specializing in group theory and the representation theory of finite groups. His ''Endliche Gruppen'' ( finite groups) is an influential textbook in group theory, ...

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 The Berlin Wall (german: Berliner Mauer, ) was a guarded concrete barrier that encircled West Berlin from 1961 to 1989, separating it from East Berlin and East Germany (GDR). Construction of the Berlin Wall was commenced by the government ...

.

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 of finite simple groups, Classification theorem was completed.

References

{{SIA, mathematics Sporadic groups