Janko group J1
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- The correct title of this article is Janko group J1. It features superscript or subscript characters that are substituted or omitted because of technical limitations.
In mathematics, the smallest Janko group, J1, of order 175560, was first described by Zvonimir Janko (1965), in a paper which described the first new sporadic simple group to be discovered in over a century and which launched the modern theory of sporadic simple groups.
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[edit] Properties
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 the smallest of the 6 sporadic simple groups called the pariahs, because they are not found within the Monster group. J1 is contained in the O'Nan group as the subgroup of elements fixed by an outer automorphism of order 2.
[edit] Construction
Janko found a modular representation in terms of 7 × 7 orthogonal matrices in the field of eleven elements, with generators given by
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).
There is also a pair of generators a, b such that
- a2=b3=(ab)7=(abab−1)19=1
J1 is thus a Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group.
[edit] Maximal subgroups
Janko (1966) enumerated all 7 conjugacy classes of maximal subgroups (see also the Atlas webpages cited below). 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.
Here is a complete list of the maximal subgroups.
| 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.
[edit] References
- Zvonimir Janko, A new finite simple group with abelian Sylow subgroups, Proc. Nat. 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.
- Robert A. Wilson, Is J1 a subgroup of the monster?, Bull. London Math. Soc. 18, no. 4 (1986), 349-350.
- Atlas of Finite Group Representations: J1 version 2
- Atlas of Finite Group Representations: J1 version 3
