Janko group

Last updated April 11, 2019

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.

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.

In the area of modern algebra known as group theory, the Janko groupJ₁ is a sporadic simple group of order

In the area of modern algebra known as group theory, the Janko groupJ₂ or the Hall-Janko groupHJ is a sporadic simple group of order

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.

In group theory, a topic in abstract algebra, the Mathieu groups are the five sporadic simple groups M₁₁, M₁₂, M₂₂, M₂₃ and M₂₄ introduced by Mathieu. They are multiply transitive permutation groups on 11, 12, 22, 23 or 24 objects. They were the first sporadic groups to be discovered.

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.

In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four broad classes described below. These groups can be seen as the basic building blocks of all finite groups, in a way reminiscent of the way the prime numbers are the basic building blocks of the natural numbers. The Jordan–Hölder theorem is a more precise way of stating this fact about finite groups. However, a significant difference from integer factorization is that such "building blocks" do not necessarily determine a unique group, since there might be many non-isomorphic groups with the same composition series or, put in another way, the extension problem does not have a unique solution.

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In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding quotient group. This process can be repeated, and for finite groups one eventually arrives at uniquely determined simple groups, by the Jordan–Hölder theorem.

In group theory, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple 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 (Conway 1968, 1969).

In the area of modern algebra known as group theory, the Lyons groupLy or Lyons-Sims groupLyS is a sporadic simple group of order

Zvonimir Janko (born 26 November 1932) is a Croatian mathematician who is 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 were then called the Mathieu groups. It was after 90 years of the discovery of the last Mathieu group that Zvonimir Janko constructed a new sporadic simple group in 1964. In his honour, this group is now called J₁. This discovery launched the modern theory of sporadic groups and it was an important milestone in the classification of finite simple groups.

Michio Suzuki was a Japanese mathematician who studied group theory.

Many branches of mathematics study objects of a given type and prove a classification theorem. A common theme is that the classification results in a number of series of objects and a finite number of exceptions that do not fit into any series. These are known as exceptional objects. Frequently these exceptional objects play a further and important role in the subject. Furthermore, the exceptional objects in one branch of mathematics are often related to the exceptional objects in others.

In the area of modern algebra known as group theory, the Janko groupJ₃ or the Higman-Janko-McKay groupHJM is a sporadic simple group of order

In the area of modern algebra known as group theory, the Janko groupJ₄ is a sporadic simple group of order

In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. More precisely it is a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an abelian group of odd order normalizing a non-trivial 2-subgroup of G. When G is a group of Lie type of characteristic 2 type, the width is usually the rank.

In the area of modern algebra known as group theory, the Mathieu groupM₁₁ is a sporadic simple group of order

In the area of modern algebra known as group theory, the Mathieu groupM₂₂ is a sporadic simple group of order

In the area of modern algebra known as group theory, the Mathieu groupM₂₃ is a sporadic simple group of order

In the area of modern algebra known as group theory, the Mathieu groupM₂₄ is a sporadic simple group of order

In the mathematical branch of moonshine theory, a supersingular prime is a prime number that divides the order of the Monster group M, which is the largest sporadic simple group. There are precisely fifteen supersingular prime numbers: the first eleven primes, as well as 41, 47, 59, and 71.

In the mathematical classification of finite simple groups, a thin group is a finite group such that for every odd prime number p, the Sylow p-subgroups of the 2-local subgroups are cyclic. Informally, these are the groups that resemble rank 1 groups of Lie type over a finite field of characteristic 2.

References

↑ Dieter Held, Die Klassifikation der endlichen einfachen Gruppen Archived 2013-06-26 at the Wayback Machine . (the classification of the finite simple groups), Forschungsmagazin der Johannes Gutenberg-Universität Mainz 1/86
↑ 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."

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[1] Dieter Held, Die Klassifikation der endlichen einfachen Gruppen Archived 2013-06-26 at the Wayback Machine . (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."