Janko group J3
Sporadic simple group
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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 = 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 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.
The degrees of irreducible representations of the Janko group J3 are 1, 85, 85, 323, 323, 324, ... (sequence A003906 in the OEIS).
Constructions
Using matrices
J3 can be constructed by many different generators.[2] Two from the ATLAS list are 18×18 matrices over the finite field of order 9, with matrix multiplication carried out with finite field arithmetic:
and
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:
where is the Frobenius automorphism of order 4, and is the unique 17-cycle that sends
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.}](http://wikimedia.org/api/rest_v1/media/math/render/svg/58d1b22df8960eecb0fa0cb95e013899c64d4e30)
A presentation for J3 in terms of (different) generators a, b, c, d is
Maximal subgroups
Finkelstein & Rudvalis (1974) found the 9 conjugacy classes of maximal subgroups of J3 as follows:
| 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 |