Janko group J1

For general background and history of the Janko sporadic groups, see Janko group.

In the area of modern algebra known as group theory, the Janko group J₁ is a sporadic simple group of order

   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. Thus it is one of the 6 sporadic groups called the pariahs.

Properties

The smallest faithful complex representation of J₁ has dimension 56. J₁ can be characterized abstractly as the unique simple group with abelian 2-Sylow subgroups and with an involution whose centralizer is isomorphic to the direct product of the group of order two and the alternating group A₅ of order 60, which is to say, the rotational icosahedral group. That was Janko's original conception of the group. In fact Janko and Thompson were investigating groups similar to the Ree groups ²G₂(3²ⁿ⁺¹), and showed that if a simple group G has abelian Sylow 2-subgroups and a centralizer of an involution of the form Z/2Z×PSL₂(q) for q a prime power at least 3, then either q is a power of 3 and G has the same order as a Ree group (it was later shown that G must be a Ree group in this case) or q is 4 or 5. Note that PSL₂(4)=PSL₂(5)=A₅. This last exceptional case led to the Janko group J₁.

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.

J₁ is the unique finite group G with the property that for C any nontrivial conjugacy class, every element of G is equal to xy for some x, y in C.

Constructions

Modulo 11 representation

Janko found a modular representation in terms of 7 × 7 orthogonal matrices in the field of eleven elements, with generators given by

${\mathbf {Y} }=\left({\begin{matrix}0&1&0&0&0&0&0\\0&0&1&0&0&0&0\\0&0&0&1&0&0&0\\0&0&0&0&1&0&0\\0&0&0&0&0&1&0\\0&0&0&0&0&0&1\\1&0&0&0&0&0&0\end{matrix}}\right)$

and

${\mathbf {Z} }=\left({\begin{matrix}-3&+2&-1&-1&-3&-1&-3\\-2&+1&+1&+3&+1&+3&+3\\-1&-1&-3&-1&-3&-3&+2\\-1&-3&-1&-3&-3&+2&-1\\-3&-1&-3&-3&+2&-1&-1\\+1&+3&+3&-2&+1&+1&+3\\+3&+3&-2&+1&+1&+3&+1\end{matrix}}\right).$

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,...,t^X. 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:

• 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 tⁱt^jtⁱt^jtⁱ 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 tⁱt^jt^ktⁱt^j is the unique element of order 6 in PSL₂(11) that sends i to j, j to k, k back to i, so (tⁱt^jt^ktⁱt^j)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 Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group.

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 D_2n 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: Société Mathématique de France, pp. 293–307, 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. doi:10.1112/S1461157000000930.
• 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) doi:10.1016/0021-8693(66)90010-X
• Zvonimir Janko and John G. Thompson, On a Class of Finite Simple Groups of Ree, Journal of Algebra, 4 (1966), 274-292.

External links

• MathWorld: Janko Groups
• Atlas of Finite Group Representations: J₁ version 2
• Atlas of Finite Group Representations: J₁ version 3