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 groupJ₃ or the Higman-Janko-McKay groupHJM is a sporadic simple group of order

   50,232,960 = 2⁷ ·3⁵ ·5 ·17 ·19.

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 GrahamHigman 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 of 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:

Maximal subgroups of J₃

No.	Structure	Order	Index	Comments
1	L2(16):2	8,160 = 25·3·5·17	6,156 = 22·34·19
2,3	L2(19)	3,420 = 22·32·5·19	14,688 = 25·33·17	two classes, fused by an outer automorphism
4	24: (3 × A5)	2,880 = 26·32·5	17,442 = 2·33·17·19
5	L2(17)	2,448 = 24·32·17	20,520 = 23·33·5·19	centralizer of an outer automorphism of order 2
6	(3 × A6):22	2,160 = 24·33·5	23,256 = 23·32·17·19	normalizer of a subgroup of order 3 (class 3A)
7	32+1+2:8	1,944 = 23·35	25,840 = 24·5·17·19	normalizer of a Sylow 3-subgroup
8	21+4  –:A5	1,920 = 27·3·5	26,163 = 34·17·19	centralizer of involution
9	22+4: (3 × S3)	1,152 = 27·32	43,605 = 33·5·17·19

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. MR   0244371
• 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