Janko group J1

For general background and history of the Janko sporadic groups, see Janko group.

In the area of modern algebra known as group theory, the Janko groupJ₁ is a sporadic simple group of order

${\displaystyle 175,560=2^{3}\cdot 3\cdot 5\cdot 7\cdot 11\cdot 19\approx 2\times 10^{5}}$.

History

${\displaystyle J_{1}}$ 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 ${\displaystyle J_{1}}$ cannot be a subgroup of the monster group. ^[1] Thus it is one of the 6 sporadic groups called the pariahs.

Properties

The smallest faithful complex representation of ${\displaystyle J_{1}}$ has dimension 56. ^[2] ${\displaystyle J_{1}}$ 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 ${\displaystyle A_{5}}$ 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 ${\displaystyle ^{2}G_{2}(3^{2n+1})}$, and showed that if a simple group ${\displaystyle G}$ has abelian Sylow 2-subgroups and a centralizer of an involution of the form ${\displaystyle \mathbb {Z} /2\mathbb {Z} \times \operatorname {PSL} _{2}(q)}$ for ${\displaystyle q}$ a prime power at least 3, then either ${\displaystyle q}$ is a power of 3 and ${\displaystyle G}$ has the same order as a Ree group (it was later shown that ${\displaystyle G}$ must be a Ree group in this case) or ${\displaystyle q}$ is 4 or 5. Note that ${\displaystyle \operatorname {PSL} _{2}(4)=\operatorname {PSL} _{2}(5)=A_{5}}$. This last exceptional case led to the Janko group ${\displaystyle J_{1}}$.

${\displaystyle J_{1}}$ has no outer automorphisms and its Schur multiplier is trivial.

${\displaystyle J_{1}}$ is contained in the O'Nan group as the subgroup of elements fixed by an outer automorphism of order 2.

${\displaystyle J_{1}}$ is the unique finite group ${\displaystyle G}$ with the property that for ${\displaystyle C}$ any nontrivial conjugacy class, every element of ${\displaystyle G}$ is equal to ${\displaystyle xy}$ for some ${\displaystyle x,y}$ in ${\displaystyle C}$. ^[3]

Constructions

Modulo 11 representation

Janko found a modular representation in terms of ${\displaystyle 7\times 7}$ orthogonal matrices in the field of eleven elements, with generators given by

${\displaystyle Y=\left({\begin{matrix}0&1&0&0&0&0&0\\0&0&1&0&0&0&0\\0&0&0&1&0&0&0\\0&0&0&0&1&0&0\\0&0&0&0&0&1&0\\0&0&0&0&0&0&1\\1&0&0&0&0&0&0\end{matrix}}\right)}$

and

${\displaystyle Z=\left({\begin{matrix}-3&+2&-1&-1&-3&-1&-3\\-2&+1&+1&+3&+1&+3&+3\\-1&-1&-3&-1&-3&-3&+2\\-1&-3&-1&-3&-3&+2&-1\\-3&-1&-3&-3&+2&-1&-1\\+1&+3&+3&-2&+1&+1&+3\\+3&+3&-2&+1&+1&+3&+1\end{matrix}}\right).}$

${\displaystyle Y}$ has order 7 and ${\displaystyle Z}$ 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

${\displaystyle J_{1}}$ is the automorphism group of the Livingstone graph, a distance-transitive graph with 266 vertices and 1463 edges. The stabilizer of a vertex is ${\displaystyle \operatorname {PSL} _{2}(11)}$, and the stabilizer of an edge is ${\displaystyle 2\times A_{5}}$.

This permutation representation can be constructed implicitly by starting with the subgroup ${\displaystyle \operatorname {PSL} _{2}(11)}$ and adjoining 11 involutions ${\displaystyle t^{0},\dots ,t^{X}}$. ${\displaystyle \operatorname {PSL} _{2}(11)}$ 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 ${\displaystyle J_{1}}$: ^[4]

• Given points ${\displaystyle i}$ and ${\displaystyle j}$, there are 2 lines containing both ${\displaystyle i}$ and ${\displaystyle j}$, and 3 points lie on neither of these lines: the product ${\displaystyle t^{i}t^{j}t^{i}t^{j}t^{i}}$ is the unique involution in ${\displaystyle \operatorname {PSL} _{2}(11)}$ that fixes those 3 points.
• Given points ${\displaystyle i}$, ${\displaystyle j}$, and ${\displaystyle k}$ that do not lie in a common line, the product ${\displaystyle t^{i}t^{j}t^{k}t^{i}t^{j}}$ is the unique element of order 6 in ${\displaystyle \operatorname {PSL} _{2}(11)}$ that sends ${\displaystyle i}$ to ${\displaystyle j}$, ${\displaystyle j}$ to ${\displaystyle k}$, ${\displaystyle k}$ back to ${\displaystyle i}$, so ${\displaystyle (t^{i}t^{j}t^{k}t^{i}t^{j})^{j}}$ is the unique involution that fixes these 3 points.

Presentation

There is also a pair of generators ${\displaystyle a}$, ${\displaystyle b}$ such that

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

${\displaystyle J_{1}}$ 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 ${\displaystyle J_{1}}$ shown in the table. Maximal simple subgroups of order 660 afford ${\displaystyle J_{1}}$ a permutation representation of degree 266. He found that there are 2 conjugacy classes of subgroups isomorphic to the alternating group ${\displaystyle A_{5}}$, both found in the simple subgroups of order 660. ${\displaystyle J_{1}}$ has non-abelian simple proper subgroups of only 2 isomorphism types.

Maximal subgroups of J₁

No.	Structure	Order	Index	Description
1	L2(11)	660 = 22·3·5·11	266 = 2·7·19	fixes point in smallest permutation representation
2	23:7:3	168 = 23·3·7	1,045 = 5·11·19	normalizer of Sylow 2-subgroup
3	2×A5	120 = 23·3·5	1,463 = 7·11·19	centralizer of involution
4	19:6	114 = 2·3·19	1,540 = 22·5·7·11	normalizer of Sylow 19-subgroup
5	11:10	110 = 2·5·11	1,596 = 22·3·7·19	normalizer of Sylow 11-subgroup
6	D6×D10	60 = 22·3·5	2,926 = 2·7·11·19	normalizer of Sylow 3-subgroup and Sylow 5-subgroup
7	7:6	42 = 2·3·7	4,180 = 22·5·11·19	normalizer of Sylow 7-subgroup

In this table, ${\displaystyle D_{2n}}$ is the dihedral group of order ${\displaystyle 2n}$.

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

1. Wilson (1986). "Is J₁ a subgroup of the Monster?". Bulletin of the London Mathematical Society. 18 (4): 349–350. doi: 10.1112/blms/18.4.349 .
2. Jansen (2005), p.123
3. Arad, Z.; Fisman, E. (1985), p.7
4. Curtis, R. T. (1993), "Symmetric Presentations II: The Janko Group J₁", Journal of the London Mathematical Society (2): 294–308, doi:10.1112/jlms/s2-47.2.294, ISSN   0024-6107

• Chevalley, Claude (1995) [1967], "Le groupe de Janko", Séminaire Bourbaki, Vol. 10, Paris: Société Mathématique de France, pp. 293–307, MR   1610425
• 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.
• Christoph, Jansen (2005). "The minimal degrees of faithful representations of the sporadic simple groups and their covering groups". LMS Journal of Computation and Mathematics. 8: 123. doi: 10.1112/S1461157000000930 .
• 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) 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.

External links

• MathWorld: Janko Groups
• Atlas of Finite Group Representations: J₁ version 2
• Atlas of Finite Group Representations: J₁ version 3