Janko group J3 explained

In the area of modern algebra known as group theory, the Janko group J₃ or the Higman-Janko-McKay group HJM is a sporadic simple group of order

   2⁷3⁵51719 = 50232960.

History and properties

J₃ is one of the 26 Sporadic groups and was predicted by Zvonimir Janko in 1969 as one of two new simple groups having 2¹⁺⁴:A₅ as a centralizer of an involution (the other is the Janko group J₂). J₃ was shown to exist by .

In 1982 R. L. Griess showed that J₃ cannot be a subquotient of the monster group.^[1] Thus it is one of the 6 sporadic groups called the pariahs.

J₃ has an outer automorphism group of order 2 and a Schur multiplier of order 3, and its triple cover has a unitary 9-dimensional representation over the finite field with 4 elements. constructed it via an underlying geometry. It has a modular representation of dimension eighteen over the finite field with 9 elements.It has a complex projective representation of dimension eighteen.

Constructions

Using matrices

J3 can be constructed by many different generators.^[2] Two from the ATLAS list are 18x18 matrices over the finite field of order 9, with matrix multiplication carried out with finite field arithmetic:

\left( \begin{matrix} 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 \\ 3 & 7 & 4 & 8 & 4 & 8 & 1 & 5 & 5 & 1 & 2 & 0 & 8 & 6 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 \\ 4 & 8 & 6 & 2 & 4 & 8 & 0 & 4 & 0 & 8 & 4 & 5 & 0 & 8 & 1 & 1 & 8 & 5 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 \\ \end{matrix} \right)

and

\left( \begin{matrix} 4 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 4 & 4 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 \\ 2 & 7 & 4 & 5 & 7 & 4 & 8 & 5 & 6 & 7 & 2 & 2 & 8 & 8 & 0 & 0 & 5 & 0 \\ 4 & 7 & 5 & 8 & 6 & 1 & 1 & 6 & 5 & 3 & 8 & 7 & 5 & 0 & 8 & 8 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 8 & 2 & 5 & 5 & 7 & 2 & 8 & 1 & 5 & 5 & 7 & 8 & 6 & 0 & 0 & 7 & 3 & 8 \\ \end{matrix} \right)

Using the subgroup PSL(2,16)

The automorphism group J₃:2 can be constructed by starting with the subgroup PSL(2,16):4 and adjoining 120 involutions, which are identified with the Sylow 17-subgroups. Note that these 120 involutions are outer elements of J₃:2. One then defines the following relation:

\left(\begin{matrix} 1 & 1 \\ 1 & 0 \end{matrix}\sigma t _{(\nu,\nu7)} \right) ^{5=1}

where

\sigma is the Frobenius automorphism or order 4, and t _{(\nu,\nu7)} is the unique 17-cycle that sendsinf ty → 0 → 1 → 7

Curtis showed, using a computer, that this relation is sufficient to define J₃:2.

Using a presentation

In terms of generators a, b, c, and d its automorphism group J₃:2 can be presented as

a ^{17} = b ^{8} = a ^{ba} ^{-2} = c ^{2} = b ^{cb} ^{3} = (abc) ^{4} = (ac) ^{17} = d ^{2} = [d, a] = [d, b] = (a ^{3b} ^{-3} cd) ^{5} = 1 .

A presentation for J₃ in terms of (different) generators a, b, c, d is

a ^{19} = b ^{9} = a ^{ba} ^{2} = c ^{2} = d ^{2} = (bc) ^{2} = (bd) ^{2} = (ac) ^{3} = (ad) ^{3} = (a ^{2ca} ^{-3} d) ^{3} = 1 .

Maximal subgroups

found the 9 conjugacy classes of maximal subgroups of J₃ as follows:

• PSL(2,16):2, order 8160
• PSL(2,19), order 3420
• PSL(2,19), conjugate to preceding class in J₃:2
• 2⁴: (3 × A₅), order 2880
• PSL(2,17), order 2448
• (3 × A₆):2₂, order 2160 - normalizer of subgroup of order 3
• 3²⁺¹⁺²:8, order 1944 - normalizer of Sylow 3-subgroup
• 2¹⁺⁴:A₅, order 1920 - centralizer of involution
• 2²⁺⁴: (3 × S₃), order 1152

References

• • R. L. Griess, Jr., The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102. p. 93: proof that J₃ is a pariah.
  • Z. Janko, Some new finite simple groups of finite order, 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 pp. 25–64 Academic Press, London, and in The theory of finite groups (Edited by Brauer and Sah) p. 63-64, Benjamin, 1969.
• Richard . Weiss . A Geometric Construction of Janko's Group J₃ . Mathematische Zeitschrift . 179 . 91–95 . 1982. 179 . 10.1007/BF01173917 .

External links

• MathWorld: Janko Groups
• Atlas of Finite Group Representations: J₃ version 2
• Atlas of Finite Group Representations: J₃ version 3

Notes and References

1. Griess (1982): p. 93: proof that J₃ is a pariah.
2. http://brauer.maths.qmul.ac.uk/Atlas/spor/J3/ ATLAS page on J3