- Janko group J1
In
mathematics , the smallestJanko group , J1, of order 175560, was first described byZvonimir Janko (1965), in a paper which described the first new sporadic simple group to be discovered in over a century and which launched the modern theory ofsporadic simple group s.Properties
J1 can be characterized abstractly as the unique
simple group with abelian 2-Sylow subgroups and with aninvolution whosecentralizer is isomorphic to thedirect product of the group of order two and thealternating 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 theRee group s 2"G"2(32"n"+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×"PSL"2("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 "PSL"2("4")="PSL"2("5")="A"5. 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 theO'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 ( egin{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} ight )and:mathbf Z} = left ( egin{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} ight ).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 "G"2(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)19=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 thealternating 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.
The notation "A"."B" means a group with a normal subgroup "A" with quotient "B", and "D"2"n" is the dihedral group of order 2"n".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 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
* 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) 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.
* Robert A. Wilson, "Is J1 a subgroup of the monster?", Bull. London Math. Soc. 18, no. 4 (1986), 349-350.
* [http://web.mat.bham.ac.uk/atlas/v2.0/spor/J1/ Atlas of Finite Group Representations: "J"1] version 2
* [http://brauer.maths.qmul.ac.uk/Atlas/v3/spor/J1/ Atlas of Finite Group Representations: "J"1] version 3
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