Weitzenböck's inequality

In mathematics, the smallest Janko group, J₁, is a simple sporadic group of order $175560=19\cdot 11\cdot 7\cdot 5\cdot 3\cdot 2^3$. 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

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 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 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

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

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

and

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

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

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 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

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• 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) Electronic Instrument Positions Staff (Standard ) Cameron from Clarence Creek, usually spends time with hobbies and interests which include knotting, property developers in singapore apartment For sale and boomerangs. Has enrolled in a world contiki journey. Is extremely thrilled specifically about visiting .
• 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