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
editJ4 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
editThe 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.
Presentation
editIt 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
editKleidman & 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.