Janko group J3

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In mathematics, the third Janko group J₃, also known as the Higman-Janko-McKay group, is a finite simple sporadic group of order 50232960. Evidence for its existence was uncovered by Zvonimir Janko, and it was shown to exist by Graham Higman and John McKay. In terms of generators a, b, c, and d its automorphism group J₃:2 can be presented as a¹⁷ = b⁸ = a^ba ^{− 2} = c² = b^cb³ = (abc)⁴ = (ac)¹⁷ = d² = [d,a] = [d,b] = (a³b ^{− 3}cd)⁵ = 1. A presentation for J₃ in terms of (different) generators a, b, c, d is a¹⁹ = b⁹ = a^ba² = c² = d² = (bc)² = (bd)² = (ac)³ = (ad)³ = (a²ca ^{− 3}d)³ = 1. It 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 1152

Janko predicted both J₃ and J₂ as simple groups having 2¹⁺⁴:A₅ as a centralizer of an involution.

[edit] 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.MR0244371
• 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, MR0246955, ISSN 0024-6093 
• 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.
• Atlas of Finite Group Representations: J₃