J3 is one of the 26 Sporadic groups and was predicted by Zvonimir Janko in 1969 as one of two new simple groups having 21+4:A5 as a centralizer of an involution (the other is the Janko group J2).
J3 was shown to exist by Graham Higman and John McKay (1969).
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.
It has a complex projective representation of dimension eighteen.
Presentationsedit
In terms of generators a, b, c, and d its automorphism group J3:2 can be presented as
A presentation for J3 in terms of (different) generators a, b, c, d is
Constructionsedit
J3 can be constructed by many different generators.[2] Two from the ATLAS list are 18x18 matrices over the finite field of order 9, with matrix multiplication carried out with finite field arithmetic:
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.MR0244371
Weiss, Richard (1982). "A Geometric Construction of Janko's Group J3". Mathematische Zeitschrift (179): 91–95.
External linksedit
MathWorld: Janko Groups
Atlas of Finite Group Representations: J3 version 2
Atlas of Finite Group Representations: J3 version 3