| Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as group theory, the Janko groupJ4 is a sporadic simple group of order
Contents
- 86,775,571,046,077,562,880
- = 221 ·33 ·5 ·7 ·113 ·23 ·29 ·31 ·37 ·43
- ≈ 9×1019.
| Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as group theory, the Janko groupJ4 is a sporadic simple group of order
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.
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.
It has a presentation in terms of three generators a, b, and c as
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.
Kleidman & Wilson (1988) found the 13 conjugacy classes of maximal subgroups of J4 which are listed in the table below.
| No. | Structure | Order | Comments |
|---|---|---|---|
| 1 | 211:M24 | 501,397,585,920 = 221.33.5.7.11.23 | 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 | 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 | |
| 4 | 23+12 · (S5 × L3(2)) | 660,602,880 = 221.32.5.7 | contains a Sylow 2-subgroup |
| 5 | U3(11):2 | 141,831,360 = 26.32.5.113.37 | |
| 6 | M22:2 | 887,040 = 28.32.5.7.11 | |
| 7 | 111+2 +:(5 × GL(2,3)) | 319,440 = 24.3.5.113 | normalizer of a Sylow 11-subgroup |
| 8 | L2(32):5 | 163,680 = 25.3.5.11.31 | |
| 9 | PGL(2,23) | 12,144 = 24.3.11.23 | |
| 10 | U3(3) | 6,048 = 25.33.7 | contains a Sylow 3-subgroup |
| 11 | 29:28 | 812 = 22.7.29 | Frobenius group; normalizer of a Sylow 29-subgroup |
| 12 | 43:14 | 602 = 2.7.43 | Frobenius group; normalizer of a Sylow 43-subgroup |
| 13 | 37:12 | 444 = 22.3.37 | 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.
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808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000
= 246 · 320 · 59 · 76 · 112 · 133 · 17 · 19 · 23 · 29 · 31 · 41 · 47 · 59 · 71
≈ 8×1053.
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