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.
The smallest faithful complex representation of J1 has dimension 56.[2]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 groups2G2(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.
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).
Janko (1966) 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.
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 orderedit
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
Referencesedit
^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.
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 linksedit
MathWorld: Janko Groups
version 2
Atlas of Finite Group Representations: J1 version 3
December 02, 2023
janko, group, general, background, history, janko, sporadic, groups, janko, group, area, modern, algebra, known, group, theory, janko, group, sporadic, simple, group, order, 175560, contents, history, properties, construction, maximal, subgroups, number, eleme. 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 group J1 is a sporadic simple group of order 23 3 5 7 11 19 175560 2 105 Contents 1 History 2 Properties 3 Construction 4 Maximal subgroups 5 Number of elements of each order 6 References 7 External linksHistory editJ1 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 1 Thus it is one of the 6 sporadic groups called the pariahs Properties editThe smallest faithful complex representation of J1 has dimension 56 2 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 is the automorphism group of the Livingstone graph a distance transitive graph with 266 vertices and 1463 edges J1 has no outer automorphisms and its Schur multiplier is trivial J1 is contained in the O Nan group as the subgroup of elements fixed by an outer automorphism of order 2 Construction editJanko found a modular representation in terms of 7 7 orthogonal matrices in the field of eleven elements with generators given by Y 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 displaystyle mathbf Y left begin matrix 0 amp 1 amp 0 amp 0 amp 0 amp 0 amp 0 0 amp 0 amp 1 amp 0 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 1 amp 0 amp 0 amp 0 0 amp 0 amp 0 amp 0 amp 1 amp 0 amp 0 0 amp 0 amp 0 amp 0 amp 0 amp 1 amp 0 0 amp 0 amp 0 amp 0 amp 0 amp 0 amp 1 1 amp 0 amp 0 amp 0 amp 0 amp 0 amp 0 end matrix right nbsp and Z 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 displaystyle mathbf Z left begin matrix 3 amp 2 amp 1 amp 1 amp 3 amp 1 amp 3 2 amp 1 amp 1 amp 3 amp 1 amp 3 amp 3 1 amp 1 amp 3 amp 1 amp 3 amp 3 amp 2 1 amp 3 amp 1 amp 3 amp 3 amp 2 amp 1 3 amp 1 amp 3 amp 3 amp 2 amp 1 amp 1 1 amp 3 amp 3 amp 2 amp 1 amp 1 amp 3 3 amp 3 amp 2 amp 1 amp 1 amp 3 amp 1 end matrix right nbsp 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 1J1 is thus a Hurwitz group a finite homomorphic image of the 2 3 7 triangle group Maximal subgroups editJanko 1966 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 Structure Order Index DescriptionPSL2 11 660 266 Fixes point in smallest permutation representation23 7 3 168 1045 Normalizer of Sylow 2 subgroup2 A5 120 1463 Centralizer of involution19 6 114 1540 Normalizer of Sylow 19 subgroup11 10 110 1596 Normalizer of Sylow 11 subgroupD6 D10 60 2926 Normalizer of Sylow 3 subgroup and Sylow 5 subgroup7 6 42 4180 Normalizer of Sylow 7 subgroupThe 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 editThe greatest order of any element of the group is 19 The conjugacy class orders and sizes are found in the ATLAS Order No elements Conjugacy1 1 1 1 1 class2 2 1463 7 11 19 1 class3 3 5852 22 7 11 19 1 class5 5 11704 23 7 11 19 2 classes power equivalent6 2 3 29260 22 5 7 11 19 1 class7 7 25080 23 3 5 11 19 1 class10 2 5 35112 23 3 7 11 19 2 classes power equivalent11 11 15960 23 3 5 7 19 1 class15 3 5 23408 24 7 11 19 2 classes power equivalent19 19 27720 23 32 5 7 11 3 classes power equivalentReferences edit 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 Jansen 2005 p 123 Chevalley Claude 1995 1967 Le groupe de Janko Seminaire Bourbaki Vol 10 Paris Societe Mathematique 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 editMathWorld Janko Groups Atlas of Finite Group Representations J1 version 2 Atlas of Finite Group Representations J1 version 3 Retrieved from https en wikipedia org w index php title Janko group J1 amp oldid 1170064665, wikipedia, wiki, book, books, library,