Janko group J4

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In mathematics, the fourth Janko group J4 is the sporadic finite simple group of order

221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43 (= 86775571046077562880)

whose existence was suggested by Janko (1976).

The Schur multiplier and the outer automorphism group are both trivial.

J4 is one of the 6 sporadic simple groups known as the "pariah groups" as they are not found within the Monster group. The order of the monster group is not divisible by 37 or 43.

Existence and uniqueness[edit]

Janko found J₄ by studying groups with an involution centralizer of the form 21 + 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 of dimension 112 over the finite field of two 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. Aschbacher & Segev (1991) and Ivanov (1992) gave computer-free proofs of uniqueness. Ivanov & Meierfrankenfeld (1999) and Ivanov (2004) 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.

Representations[edit]

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, with point stabilizer of the form 211M24. These points can be identified with certain "special vectors" in the 112 dimensional representation.

Presentation[edit]

It has a presentation in terms of three generators a, b, and c as

\begin{align}
a^2 &=b^3=c^2=(ab)^{23}=[a,b]^{12}=[a,bab]^5=[c,a]= \left (ababab^{-1} \right)^3 \left (abab^{-1}ab^{-1} \right)^3=\left (ab \left (abab^{-1} \right )^3 \right )^4 \\
&=\left [c,bab \left(ab^{-1} \right )^2(ab)^3 \right]= \left (bc^{bab^{-1}abab^{-1}a} \right )^3= \left ((bababab)^3 c c^{(ab)^3b(ab)^6b} \right )^2=1.
\end{align}

Maximal subgroups[edit]

Kleidman & Wilson (1988) showed that J4 has 13 conjugacy classes of maximal subgroups.

  • 211:M24 - containing Sylow 2-subgroups and Sylow 3-subgroups; also containing 211:(M22:2), centralizer of involution of class 2B
  • 21+12.3.(M22:2) - centralizer of involution of class 2A - containing Sylow 2-subgroups and Sylow 3-subgroups
  • 210:PSL(5,2)
  • 23+12.(S5 × PSL(3,2)) - containing Sylow 2-subgroups
  • U3(11):2
  • M22:2
  • 111+2:(5 × GL(2,3)) - normalizer of Sylow 11-subgroup
  • PSL(2,32):5
  • PGL(2,23)
  • U3(3) - containing Sylow 3-subgroups
  • 29:28 Frobenius group
  • 43:14 Frobenius group
  • 37:12 Frobenius group

A Sylow 3-subgroup is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3

References[edit]

External links[edit]