Janko group J1
Algebraic structure → Group theory Group theory |
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In the area of modern algebra known as
- 23 · 3 · 5 · 7 · 11 · 19 = 175560
- ≈ 2×105.
History
J1 is one of the 26 sporadic groups and was originally described by Zvonimir Janko in 1965. It is the only Janko group whose existence was proved by Janko himself and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century. Its discovery launched the modern theory of sporadic groups.
In 1986 Robert A. Wilson showed that J1 cannot be a subgroup of the monster group.[1] Thus it is one of the 6 sporadic groups called the pariahs.
Properties
The smallest faithful complex representation of J1 has dimension 56.
J1 has no outer automorphisms and its Schur multiplier is trivial.
J1 is contained in the O'Nan group as the subgroup of elements fixed by an outer automorphism of order 2.
Constructions
Modulo 11 representation
Janko found a
and
Y has order 7 and Z has order 5. Janko (1966) credited W. A. Coppel for recognizing this representation as an embedding into Dickson's simple group G2(11) (which has a 7-dimensional representation over the field with 11 elements).
Permutation representation
J1 is the automorphism group of the Livingstone graph, a distance-transitive graph with 266 vertices and 1463 edges. The stabilizer of a vertex is PSL2(11), and the stabilizer of an edge is 2×A5.
This permutation representation can be constructed implicitly by starting with the subgroup PSL2(11) and adjoining 11 involutions t0,...,tX. PSL2(11) permutes these involutions under the exceptional 11-point representation, so they may be identified with points in the Payley biplane. The following relations (combined) are sufficient to define J1:[3]
- Given points i and j, there are 2 lines containing both i and j, and 3 points lie on neither of these lines: the product titjtitjti is the unique involution in PSL2(11) that fixes those 3 points.
- Given points i, j, and k that do not lie in a common line, the product titjtktitj is the unique element of order 6 in PSL2(11) that sends i to j, j to k, k back to i, so (titjtktitj)3 is the unique involution that fixes these 3 points.
Presentation
There is also a pair of generators a, b such that
- a2=b3=(ab)7=(abab−1)10=1
J1 is thus a
Maximal subgroups
Janko (1966) found the 7 conjugacy classes of maximal subgroups of J1 shown in the table. Maximal simple subgroups of order 660 afford J1 a permutation representation of degree 266. He found that there are 2 conjugacy classes of subgroups isomorphic to the alternating group A5, both found in the simple subgroups of order 660. J1 has non-abelian simple proper subgroups of only 2 isomorphism types.
Structure | Order | Index | Description |
---|---|---|---|
PSL2(11) | 660 | 266 | Fixes point in smallest permutation representation |
23.7.3 | 168 | 1045 | Normalizer of Sylow 2-subgroup |
2×A5 | 120 | 1463 | Centralizer of involution |
19.6 | 114 | 1540 | Normalizer of Sylow 19-subgroup |
11.10 | 110 | 1596 | Normalizer of Sylow 11-subgroup |
D6×D10 | 60 | 2926 | Normalizer of Sylow 3-subgroup and Sylow 5-subgroup |
7.6 | 42 | 4180 | Normalizer of Sylow 7-subgroup |
The notation A.B means a group with a normal subgroup A with quotient B, and D2n is the dihedral group of order 2n.
Number of elements of each order
The greatest order of any element of the group is 19. The conjugacy class orders and sizes are found in the ATLAS.
Order | No. elements | Conjugacy |
---|---|---|
1 = 1 | 1 = 1 | 1 class |
2 = 2 | 1463 = 7 · 11 · 19 | 1 class |
3 = 3 | 5852 = 22 · 7 · 11 · 19 | 1 class |
5 = 5 | 11704 = 23 · 7 · 11 · 19 | 2 classes, power equivalent |
6 = 2 · 3 | 29260 = 22 · 5 · 7 · 11 · 19 | 1 class |
7 = 7 | 25080 = 23 · 3 · 5 · 11 · 19 | 1 class |
10 = 2 · 5 | 35112 = 23 · 3 · 7 · 11 · 19 | 2 classes, power equivalent |
11 = 11 | 15960 = 23 · 3 · 5 · 7 · 19 | 1 class |
15 = 3 · 5 | 23408 = 24 · 7 · 11 · 19 | 2 classes, power equivalent |
19 = 19 | 27720 = 23 · 32 · 5 · 7 · 11 | 3 classes, power equivalent |
References
- Chevalley, Claude (1995) [1967], "Le groupe de Janko", Séminaire Bourbaki, Vol. 10, Paris: MR 1610425
- Robert A. Wilson (1986). Is J1 a subgroup of the monster?, Bull. London Math. Soc. 18, no. 4 (1986), 349-350
- R. T. Curtis, (1993) Symmetric Representations II: The Janko group J1, J. London Math. Soc., 47 (2), 294-308.
- R. T. Curtis, (1996) Symmetric representation of elements of the Janko group J1, J. Symbolic Comp., 22, 201-214.
- Christoph, Jansen (2005). "The minimal degrees of faithful representations of the sporadic simple groups and their covering groups". LMS Journal of Computation and Mathematics. 8: 123. .
- Zvonimir Janko, A new finite simple group with abelian Sylow subgroups, Proc. Natl. Acad. Sci. USA 53 (1965) 657-658.
- Zvonimir Janko, A new finite simple group with abelian Sylow subgroups and its characterization, Journal of Algebra 3: 147-186, (1966)
- Zvonimir Janko and John G. Thompson, On a Class of Finite Simple Groups of Ree, Journal of Algebra, 4 (1966), 274-292.