Janko group J3
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In the area of modern algebra known as group theory, the Janko group J3 or the Higman-Janko-McKay group HJM is a sporadic simple group of order
- 27 · 35 · 5 · 17 · 19 = 50232960.
History and properties
[edit]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).
In 1982 R. L. Griess showed that J3 cannot be a subquotient of the monster group.[1] Thus it is one of the 6 sporadic groups called the pariahs.
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.
Constructions
[edit]Using matrices
[edit]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:
and
Using the subgroup PSL(2,16)
[edit]The automorphism group J3:2 can be constructed by starting with the subgroup PSL(2,16):4 and adjoining 120 involutions, which are identified with the Sylow 17-subgroups. Note that these 120 involutions are outer elements of J3:2. One then defines the following relation:
where is the Frobenius automorphism or order 4, and is the unique 17-cycle that sends
Curtis showed, using a computer, that this relation is sufficient to define J3:2.[3]
Using a presentation
[edit]In terms of generators a, b, c, and d its automorphism group J3:2 can be presented as
![{\displaystyle a^{17}=b^{8}=a^{b}a^{-2}=c^{2}=b^{c}b^{3}=(abc)^{4}=(ac)^{17}=d^{2}=[d,a]=[d,b]=(a^{3}b^{-3}cd)^{5}=1.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/58d1b22df8960eecb0fa0cb95e013899c64d4e30)
A presentation for J3 in terms of (different) generators a, b, c, d is
Maximal subgroups
[edit]Finkelstein & Rudvalis (1974) found the 9 conjugacy classes of maximal subgroups of J3 as follows:
- 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
[edit]- ^ Griess (1982): p. 93: proof that J3 is a pariah.
- ^ ATLAS page on J3
- ^ Bradley, J.D.; Curtis, R.T. (2006), "Symmetric Generationand existence of J3:2, the automorphism group of the third Janko group", Journal of Algebra, 304 (1): 256–270, doi:10.1016/j.jalgebra.2005.09.046
- Finkelstein, L.; Rudvalis, A. (1974), "The maximal subgroups of Janko's simple group of order 50,232,960", Journal of Algebra, 30 (1–3): 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 (179): 91–95. doi:10.1007/BF01173917.