Janko group J1

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In mathematics, the smallest Janko group, J1, is a simple sporadic group of order

23 ···· 11 · 19 = 175560

It was originally described by Zvonimir Janko (1965) 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.

Properties[edit]

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.

Construction[edit]

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).

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[edit]

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.

Number of elements of each order[edit]

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[edit]