Janko group J1

History

J_1 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 J_1 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 J_1 has dimension 56. J_1 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 A_5 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 ^2G_2(3^{2n+1}), and showed that if a simple group G has abelian Sylow 2-subgroups and a centralizer of an involution of the form \Z/2\Z\times\operatorname{PSL}_2(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 \operatorname{PSL}_2(4) = \operatorname{PSL}_2(5) = A_5. This last exceptional case led to the Janko group J_1.

J_1 has no outer automorphisms and its Schur multiplier is trivial.

J_1 is contained in the O'Nan group as the subgroup of elements fixed by an outer automorphism of order 2.

J_1 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\times 7 orthogonal matrices in the field of eleven elements, with generators given by :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 :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

J_1 is the automorphism group of the Livingstone graph, a distance-transitive graph with 266 vertices and 1463 edges. The stabilizer of a vertex is \operatorname{PSL}_2(11), and the stabilizer of an edge is 2\times A_5.

This permutation representation can be constructed implicitly by starting with the subgroup \operatorname{PSL}_2(11) and adjoining 11 involutions t^0, \dots, t^X. \operatorname{PSL}_2(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 J_1:

• 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 t^it^jt^it^jt^i is the unique involution in \operatorname{PSL}_2(11) that fixes those 3 points.
• Given points i, j, and k that do not lie in a common line, the product t^it^jt^kt^it^j is the unique element of order 6 in \operatorname{PSL}_2(11) that sends i to j, j to k, k back to i, so (t^it^jt^kt^it^j)^j is the unique involution that fixes these 3 points.

Presentation

There is also a pair of generators a, b such that

:a^2 = b^3 = (ab)^7 = (abab^{-1})^{10} = 1

J_1 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 J_1 shown in the table. Maximal simple subgroups of order 660 afford J_1 a permutation representation of degree 266. He found that there are 2 conjugacy classes of subgroups isomorphic to the alternating group A_5, both found in the simple subgroups of order 660. J_1 has non-abelian simple proper subgroups of only 2 isomorphism types.

In this table, 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.

References

• 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.
• 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.

References

1. Wilson. (1986). "Is J₁ a subgroup of the Monster?". Bulletin of the London Mathematical Society.
2. Jansen (2005), p.123
3. Arad, Z.; Fisman, E. (1985), p.7
4. (1993). "Symmetric Presentations II: The Janko Group ''J''₁". Journal of the London Mathematical Society.