Janko group J1


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

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.
J1 has no outer automorphisms and its Schur multiplier is trivial.

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, 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 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 or q is 4 or 5. Note that PSL2=PSL2=A5. This last exceptional case led to the Janko group J1.
J1 is contained in the O'Nan group as the subgroup of elements fixed by an outer automorphism of order 2.

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 credited W. A. Coppel for recognizing this representation as an embedding into Dickson's simple group G2.
There is also a pair of generators a, b such that
J1 is thus a Hurwitz group, a finite homomorphic image of the triangle group.

Maximal subgroups

Janko 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.
StructureOrderIndexDescription
PSL2660266Fixes point in smallest permutation representation
23.7.31681045Normalizer of Sylow 2-subgroup
2×A51201463Centralizer of involution
19.61141540Normalizer of Sylow 19-subgroup
11.101101596Normalizer of Sylow 11-subgroup
D6×D10602926Normalizer of Sylow 3-subgroup and Sylow 5-subgroup
7.6424180Normalizer 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.
OrderNo. elementsConjugacy
1 = 11 = 11 class
2 = 21463 = 7 · 11 · 191 class
3 = 35852 = 22 · 7 · 11 · 191 class
5 = 511704 = 23 · 7 · 11 · 192 classes, power equivalent
6 = 2 · 329260 = 22 · 5 · 7 · 11 · 191 class
7 = 725080 = 23 · 3 · 5 · 11 · 191 class
10 = 2 · 535112 = 23 · 3 · 7 · 11 · 192 classes, power equivalent
11 = 1115960 = 23 · 3 · 5 · 7 · 191 class
15 = 3 · 523408 = 24 · 7 · 11 · 192 classes, power equivalent
19 = 1927720 = 23 · 32 · 5 · 7 · 113 classes, power equivalent