Janko group J4
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In the area of modern algebra known as group theory, the Janko group J4 is a sporadic simple group of order
- 86,775,571,046,077,562,880
- = 221 · 33 · 5 · 7 · 113 · 23 · 29 · 31 · 37 · 43
- ≈ 9×1019.
History
[edit]J4 is one of the 26 Sporadic groups. Zvonimir Janko found J4 in 1975 by studying groups with an involution centralizer of the form 21 + 12.3.(M22:2). Its existence and uniqueness was shown using computer calculations by Simon P. Norton and others in 1980. It has a modular representation of dimension 112 over the finite field with 2 elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. Aschbacher & Segev (1991) and Ivanov (1992) gave computer-free proofs of uniqueness. Ivanov & Meierfrankenfeld (1999) and Ivanov (2004) gave a computer-free proof of existence by constructing it as an amalgams of groups 210:SL5(2) and (210:24:A8):2 over a group 210:24:A8.
The Schur multiplier and the outer automorphism group are both trivial.
Since 37 and 43 are not supersingular primes, J4 cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.
Representations
[edit]The smallest faithful complex representation has dimension 1333; there are two complex conjugate representations of this dimension. The smallest faithful representation over any field is a 112 dimensional representation over the field of 2 elements.
The smallest permutation representation is on 173067389 points and has rank 20, with point stabilizer of the form 211:M24. The points can be identified with certain "special vectors" in the 112 dimensional representation.
The degrees of irreducible representations of the Janko group J4 are 1, 1333, 1333, 299367, 299367, ... (sequence A003907 in the OEIS).
Presentation
[edit]It has a presentation in terms of three generators a, b, and c as
![{\displaystyle {\begin{aligned}a^{2}&=b^{3}=c^{2}=(ab)^{23}=[a,b]^{12}=[a,bab]^{5}=[c,a]=\left((ab)^{2}ab^{-1}\right)^{3}\left(ab(ab^{-1})^{2}\right)^{3}=\left(ab\left(abab^{-1}\right)^{3}\right)^{4}\\&=\left[c,(ba)^{2}b^{-1}ab^{-1}(ab)^{3}\right]=\left(bc^{(bab^{-1}a)^{2}}\right)^{3}=\left((bababab)^{3}cc^{(ab)^{3}b(ab)^{6}b}\right)^{2}=1.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a66e274b102008f63c9f2168382a354a03ed29c)
Alternatively, one can start with the subgroup M24 and adjoin 3975 involutions, which are identified with the trios. By adding a certain relation, certain products of commuting involutions generate the binary Golay cocode, which extends to the maximal subgroup 211:M24. Bolt, Bray, and Curtis showed, using a computer, that adding just one more relation is sufficient to define J4.
Maximal subgroups
[edit]Kleidman & Wilson (1988) found the 13 conjugacy classes of maximal subgroups of J4 which are listed in the table below.
| No. | Structure | Order | Index | Comments |
|---|---|---|---|---|
| 1 | 211:M24 | 501,397,585,920 = 221·33·5·7·11·23 |
173,067,389 = 112·29·31·37·43 |
contains a Sylow 2-subgroup and a Sylow 3-subgroup; contains the centralizer 211:(M22:2) of involution of class 2B |
| 2 | 21+12 + · 3.(M22:2) |
21,799,895,040 = 221·33·5·7·11 |
3,980,549,947 = 112·23·29·31·37·43 |
centralizer of involution of class 2A; contains a Sylow 2-subgroup and a Sylow 3-subgroup |
| 3 | 210:L5(2) | 10,239,344,640 = 220·32·5·7·31 |
8,474,719,242 = 2·3·113·23·29·37·43 |
|
| 4 | 23+12 · (S5 × L3(2)) | 660,602,880 = 221·32·5·7 |
131,358,148,251 = 3·113·23·29·31·37·43 |
contains a Sylow 2-subgroup |
| 5 | U3(11):2 | 141,831,360 = 26·32·5·113·37 |
611,822,174,208 = 215·3·7·23·29·31·43 |
|
| 6 | M22:2 | 887,040 = 28·32·5·7·11 |
97,825,995,497,472 = 213·3·112·23·29·31·37·43 |
|
| 7 | 111+2 +:(5 × GL(2,3)) |
319,440 = 24·3·5·113 |
271,649,045,348,352 = 217·32·7·23·29·31·37·43 |
normalizer of a Sylow 11-subgroup |
| 8 | L2(32):5 | 163,680 = 25·3·5·11·31 |
530,153,782,050,816 = 216·32·7·112·23·29·37·43 |
|
| 9 | PGL(2,23) | 12,144 = 24·3·11·23 |
7,145,550,975,467,520 = 217·32·5·7·112·29·31·37·43 |
|
| 10 | U3(3) | 6,048 = 25·33·7 |
14,347,812,672,962,560 = 216·5·113·23·29·31·37·43 |
contains a Sylow 3-subgroup |
| 11 | 29:28 | 812 = 22·7·29 |
106,866,466,805,514,240 = 219·33·5·113·23·31·37·43 |
Frobenius group; normalizer of a Sylow 29-subgroup |
| 12 | 43:14 | 602 = 2·7·43 |
144,145,466,853,949,440 = 220·33·5·113·23·29·31·37 |
Frobenius group; normalizer of a Sylow 43-subgroup |
| 13 | 37:12 | 444 = 22·3·37 |
195,440,475,329,003,520 = 219·32·5·7·113·23·29·31·43 |
Frobenius group; normalizer of a Sylow 37-subgroup |
A Sylow 3-subgroup of J4 is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3.
References
[edit]- Aschbacher, Michael; Segev, Yoav (1991), "The uniqueness of groups of type J4", Inventiones Mathematicae, 105 (3): 589–607, doi:10.1007/BF01232280, ISSN 0020-9910, MR 1117152, S2CID 121529060
- D.J. Benson The simple group J4, PhD Thesis, Cambridge 1981, https://web.archive.org/web/20110610013308/http://www.maths.abdn.ac.uk/~bensondj/papers/b/benson/the-simple-group-J4.pdf
- Bolt, Sean W.; Bray, John R.; Curtis, Robert T. (2007), "Symmetric Presentation of the Janko Group J4", Journal of the London Mathematical Society, 76 (3): 683–701, doi:10.1112/jlms/jdm086
- Ivanov, A. A. (1992), "A presentation for J4", Proceedings of the London Mathematical Society, Third Series, 64 (2): 369–396, doi:10.1112/plms/s3-64.2.369, ISSN 0024-6115, MR 1143229
- Ivanov, A. A.; Meierfrankenfeld, Ulrich (1999), "A computer-free construction of J4", Journal of Algebra, 219 (1): 113–172, doi:10.1006/jabr.1999.7851, ISSN 0021-8693, MR 1707666
- Ivanov, A. A. (2004). The Fourth Janko Group. Oxford Mathematical Monographs. Oxford: Clarendon Press. ISBN 0-19-852759-4.MR2124803
- Z. Janko, A new finite simple group of order 86,775,570,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroups, J. Algebra 42 (1976) 564-596. doi:10.1016/0021-8693(76)90115-0 (The title of this paper is incorrect, as the full covering group of M22 was later discovered to be larger: center of order 12, not 6.)
- Kleidman, Peter B.; Wilson, Robert A. (1988), "The maximal subgroups of J4", Proceedings of the London Mathematical Society, Third Series, 56 (3): 484–510, doi:10.1112/plms/s3-56.3.484, ISSN 0024-6115, MR 0931511
- S. P. Norton The construction of J4 in The Santa Cruz conference on finite groups (Ed. Cooperstein, Mason) Amer. Math. Soc 1980.