- Janko group J3
In
mathematics , the third Janko group J3, also known as the Higman-Janko-McKay group, is a finite simplesporadic group of order 50232960. Evidence for its existence was uncovered byZvonimir Janko , and it was shown to exist byGraham Higman and John McKay. In terms of generators a, b, c, and d its automorphism group J3:2 can be presented asA presentation for J3 in terms of (different) generators a, b, c, d isIt can also be constructed via an underlying geometry, as was done by Weiss, and has a modular representation of dimension eighteen over the finite field of nine elements, which can be expressed in terms of two generators.J3 has a
Schur multiplier of order 3, and its triple cover has a unitary 9 dimensional representation over the field with 4 elements.J3 is one of the 6 sporadic simple groups called the pariahs, because they are not found within the
Monster group .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
* 32+1+2:8, order 1944
* 21+4:A5, order 1920 - centralizer of involution
* 22+4:(3 × S3), order 1152Janko predicted both J3 and J2 as simple groups having 21+4:A5 as a centralizer of an involution.
References
*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" (Editied by Brauer and Sah) p. 63-64, Benjamin, 1969.MathSciNet|id=0244371
* | year=1969 | journal=Bull. London Math. Soc. | issn=0024-6093 | volume=1 | pages=89–94; correction p. 219
* Richard Weiss, "A Geometric Construction of Janko's Group J3", Math. Zeitung 179 pp 91-95 (1982)
*R. L. Griess , Jr., "The Friendly Giant", Inventiones Mathematicae 69 (1982), 1-102. p. 93: proof that J3 is a pariah.
* [http://web.mat.bham.ac.uk/atlas/v2.0/spor/J3/ Atlas of Finite Group Representations: "J"3]
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