Janko group J3
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Modular groups
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Infinite dimensional Lie group
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In the area of modern algebra known as group theory, the Janko group J3 is a sporadic simple group of order
- 27 · 35 · 5 · 17 · 19 = 50232960
- ≈ 5×107.
History and properties
J3 is one of the 26 Sporadic groups and is also called the Higman-Janko-McKay group or HJM. In 1969 Zvonimir Janko predicted J3 as one of two new simple groups having having 21+4:A5 as a centralizer of an involution (the other is the Janko group J2). It was shown to exist by Graham Higman and John McKay (1969).
J3 is one of the 6 sporadic simple groups called the pariahs because (Griess 1982) showed that it is not a subquotient of the monster group.
J3 has an outer automorphism group of order 2 and a Schur multiplier of order 3, and its triple cover has a unitary 9-dimensional representation over the finite field with 4 elements. Weiss (1982) constructed it via an underlying geometry. It has a modular representation of dimension eighteen over the finite field with 9 elements.
Presentations
In terms of generators a, b, c, and d its automorphism group J3:2 can be presented as
![a^{17} = b^8 = a^ba^{-2} = c^2 = b^cb^3 = (abc)^4 = (ac)^{17} = d^2 = [d, a] = [d, b] = (a^3b^{-3}cd)^5 = 1.](../I/m/8075d6efb6e6ab1414ee30cfe30e3d4d.png)
A presentation for J3 in terms of (different) generators a, b, c, d is
Maximal subgroups
Finkelstein & Rudvalis (1974) showed that J3 has 9 conjugacy classes of maximal subgroups:
- PSL(2,16):2, order 8160
- PSL(2,19), order 3420
- PSL(2,19), conjugate to preceding class in J3:2
- 24: (3 × A5), order 2880
- PSL(2,17), order 2448
- (3 × A6):22, order 2160 - normalizer of subgroup of order 3
- 32+1+2:8, order 1944 - normalizer of Sylow 3-subgroup
- 21+4:A5, order 1920 - centralizer of involution
- 22+4: (3 × S3), order 1152
References
- Finkelstein, L.; Rudvalis, A. (1974), "The maximal subgroups of Janko's simple group of order 50,232,960", Journal of Algebra 30: 122–143, doi:10.1016/0021-8693(74)90196-3, ISSN 0021-8693, MR 0354846
- R. L. Griess, Jr., The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102. p. 93: proof that J3 is a pariah.
- Higman, Graham; McKay, John (1969), "On Janko's simple group of order 50,232,960", Bull. London Math. Soc. 1: 89–94; correction p. 219, doi:10.1112/blms/1.1.89, MR 0246955
- Z. Janko, Some new finite simple groups of finite order, 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 pp. 25-64 Academic Press, London, and in The theory of finite groups (Edited by Brauer and Sah) p. 63-64, Benjamin, 1969.MR 0244371
- Richard Weiss, "A Geometric Construction of Janko's Group J3", Math. Zeitschrift 179 pp 91-95 (1982)