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 group ''J₄'' is a

sporadic simple group In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups. A simpl ...

order Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...

: 86,775,571,046,077,562,880 : = 2²¹3³5711³2329313743 : ≈ 9.

History

''J₄'' is one of the 26

Sporadic group In the mathematical classification of finite simple groups, there are a number of groups which do not fit into any infinite family. These are called the sporadic simple groups, or the sporadic finite groups, or just the sporadic groups. A simpl ...

s. Zvonimir Janko found J₄ in 1975 by studying groups with an involution centralizer of the form 2^{1 + 12}.3.(M₂₂:2). Its existence and uniqueness was shown using computer calculations by Simon P. Norton and others in 1980. It has a

modular representation Modular representation theory is a branch of mathematics, and is the part of representation theory that studies linear representations of finite groups over a field ''K'' of positive characteristic ''p'', necessarily a prime number. As well as h ...

of dimension 112 over the

finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...

with 2 elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. and gave computer-free proofs of uniqueness. and gave a computer-free proof of existence by constructing it as an amalgams of groups 2¹⁰:SL₅(2) and (2¹⁰:2⁴:A₈):2 over a group 2¹⁰:2⁴:A₈. The

Schur multiplier In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H_2(G, \Z) of a group ''G''. It was introduced by in his work on projective representations. Examples and properties The Schur multiplier \ope ...

and the

outer automorphism group In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has ...

are both trivial. Since 37 and 43 are not supersingular primes, ''J₄'' cannot be a

subquotient In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, thou ...

of the

monster group In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group; it has order : : = 2463205976112133171923293 ...

. Thus it is one of the 6 sporadic groups called the pariahs.

Representations

The smallest faithful complex representation has dimension 1333; there are two complex conjugate representations of this dimension. The smallest faithful representation over any field is a 112 dimensional representation over the field of 2 elements. The smallest permutation representation is on 173067389 points and has rank 20, with point stabilizer of the form 2¹¹:M₂₄. The points can be identified with certain "special vectors" in the 112 dimensional representation. The degrees of

irreducible representations In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...

of the Janko group ''J₄'' are 1, 1333, 1333, 299367, 299367, ... .

Presentation

It has a

presentation A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...

in terms of three generators a, b, and c as :

\left (bc^ \right )^3= \left ((bababab)^3 c c^ \right )^2=1. \end

Alternatively, one can start with the subgroup M₂₄ and adjoin 3975 involutions, which are identified with the

trios Trio may refer to: Music Groups * Trio (music), an ensemble of three performers, or a composition for such an ensemble ** Jazz trio, pianist, double bassist, drummer ** Minuet and trio, a form in classical music ** String trio, a group of three ...

. By adding a certain relation, certain products of commuting involutions generate the binary Golay cocode, which extends to the maximal subgroup 2¹¹:M₂₄. Bolt, Bray, and Curtis showed, using a computer, that adding just one more relation is sufficient to define ''J''₄.

Maximal subgroups

found the 13 conjugacy classes of maximal subgroups of ''J₄'' which are listed in the table below. A Sylow 3-subgroup of ''J₄'' is a

Heisenberg group In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form : \begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ''a, b' ...

: order 27, non-abelian, all non-trivial elements of order 3.

References

* *D.J. Benson ''The simple group J₄'', PhD Thesis, Cambridge 1981, https://web.archive.org/web/20110610013308/http://www.maths.abdn.ac.uk/~bensondj/papers/b/benson/the-simple-group-J4.pdf * * * * *Z. Janko, ''A new finite simple group of order 86,775,570,046,077,562,880 which possesses M₂₄ and the full covering group of M₂₂ as subgroups'', J. Algebra 42 (1976) 564-596. (The title of this paper is incorrect, as the full covering group of M₂₂ was later discovered to be larger: center of order 12, not 6.) * *S. P. Norton ''The construction of J₄'' in ''The Santa Cruz conference on finite groups'' (Ed. Cooperstein, Mason) Amer. Math. Soc 1980.

External links

MathWorld: Janko Groups

Atlas of Finite Group Representations: ''J''₄
version 2
Atlas of Finite Group Representations: ''J''₄
version 3 {{DEFAULTSORT:Janko group J3 Sporadic groups