Janko group J3

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

   50,232,960 = 2⁷Template:·3⁵Template:·5Template:·17Template:·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 Template:Harvs.

In 1982 R. L. Griess showed that J₃ cannot be a subquotient of the monster group. 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. Template:Harvtxt 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. Two from the ATLAS list are 18x18 matrices over the finite field of order 9, with matrix multiplication carried out with finite field arithmetic:

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

and

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

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:

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

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

$\mathrm{\infty }\to 0\to 1\to 7$

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

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

Maximal subgroups

Template:Harvtxt 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	2Template:Su: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

• Template:Citation
• R. L. Griess, Jr., The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102. p. 93: proof that J₃ is a pariah.
• Template:Citation
• 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.Template:MathSciNet
• Template:Cite journal

External links

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