Janko group J3

For general background and history of the Janko sporadic groups, see Janko group.

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

   2^7 · 3^5 · 5 · 17 · 19 = 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 2^1+4:A₅ as a centralizer of an involution (the other is the Janko group J₂). J₃ was shown to exist by Graham Higman and John McKay (1969).

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. Weiss (1982) 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:

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

and

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

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:

${\displaystyle \left({\begin{matrix}1&1\\1&0\end{matrix}}\sigma t_{(\nu ,\nu 7)}\right)^{5}=1}$

where ${\displaystyle \sigma }$ is the Frobenius automorphism or order 4, and ${\displaystyle t_{(\nu ,\nu 7)}}$ is the unique 17-cycle that sends

${\displaystyle \infty \rightarrow 0\rightarrow 1\rightarrow 7}$

Curtis showed, using a computer, that this relation is sufficient to define J₃:2.[3]

Using a presentation

In terms of generators a, b, c, and d its automorphism group J₃:2 can be presented as ${\displaystyle a^{17}=b^{8}=a^{b}a^{-2}=c^{2}=b^{c}b^{3}=(abc)^{4}=(ac)^{17}=d^{2}=[d,a]=[d,b]=(a^{3}b^{-3}cd)^{5}=1.}$

A presentation for J₃ in terms of (different) generators a, b, c, d is ${\displaystyle a^{19}=b^{9}=a^{b}a^{2}=c^{2}=d^{2}=(bc)^{2}=(bd)^{2}=(ac)^{3}=(ad)^{3}=(a^{2}ca^{-3}d)^{3}=1.}$

Maximal subgroups

Finkelstein & Rudvalis (1974) 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
• 2^4: (3 × A₅), order 2880
• PSL(2,17), order 2448
• (3 × A₆):2₂, order 2160 - normalizer of subgroup of order 3
• 3^2+1+2:8, order 1944 - normalizer of Sylow 3-subgroup
• 2^1+4:A₅, order 1920 - centralizer of involution
• 2^2+4: (3 × S₃), order 1152

References

1. Griess (1982): p. 93: proof that J₃ is a pariah.
2. ATLAS page on J3
3. Bradley, J.D.; Curtis, R.T. (2006), "Symmetric Generationand existence of J₃:2, the automorphism group of the third Janko group", Journal of Algebra, 304 (1): 256–270, doi:10.1016/j.jalgebra.2005.09.046

• Finkelstein, L.; Rudvalis, A. (1974), "The maximal subgroups of Janko's simple group of order 50,232,960", Journal of Algebra, 30 (1–3): 122–143, doi:10.1016/0021-8693(74)90196-3, ISSN 0021-8693, MR 0354846
• R. L. Griess, Jr., The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102. p. 93: proof that J₃ is a pariah.
• Higman, Graham; McKay, John (1969), "On Janko's simple group of order 50,232,960", Bull. London Math. Soc., 1: 89–94, correction p. 219, doi:10.1112/blms/1.1.89, MR 0246955
• 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.MR0244371
• Weiss, Richard (1982). "A Geometric Construction of Janko's Group J₃". Mathematische Zeitschrift. 179 (179): 91–95. doi:10.1007/BF01173917.

External links

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