Janko group J4

J₄ is one of the 6 sporadic simple groups known as the "pariah groups" as they are not found within the Monster group. The order of the monster group is not divisible by 37 or 43.

Existence and uniqueness

Janko found J₄ by studying groups with an involution centralizer of the form 2^{1 + 12}.3.(M₂₂:2). Its existence and uniqueness was shown using computer calculations by Simon P. Norton and others in 1980. It has a modular representation of dimension 112 over the finite field of two elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. Aschbacher & Segev (1991) and Ivanov (1992) gave computer-free proofs of uniqueness. Ivanov & Meierfrankenfeld (1999) and Ivanov (2004) gave a computer-free proof of existence by constructing it as an amalgams of groups 2¹⁰:SL₅(2) and (2¹⁰:2⁴:A₈):2 over a group 2¹⁰:2⁴:A_8.

Representations

The smallest faithful complex representation has dimension 1333; there are two complex conjugate representations of this dimension. The smallest faithful representation over any field is a 112 dimensional representation over the field of 2 elements.

The smallest permutation representation is on 173067389 points, with point stabilizer of the form 2¹¹M₂₄. These points can be identified with certain "special vectors" in the 112 dimensional representation.

Presentation

It has a presentation in terms of three generators a, b, and c as

${\begin{aligned}a^{2}&=b^{3}=c^{2}=(ab)^{{23}}=[a,b]^{{12}}=[a,bab]^{5}=[c,a]=\left(ababab^{{-1}}\right)^{3}\left(abab^{{-1}}ab^{{-1}}\right)^{3}=\left(ab\left(abab^{{-1}}\right)^{3}\right)^{4}\\&=\left[c,bab\left(ab^{{-1}}\right)^{2}(ab)^{3}\right]=\left(bc^{{bab^{{-1}}abab^{{-1}}a}}\right)^{3}=\left((bababab)^{3}cc^{{(ab)^{3}b(ab)^{6}b}}\right)^{2}=1.\end{aligned}}$

Maximal subgroups

Kleidman & Wilson (1988) showed that J₄ has 13 conjugacy classes of maximal subgroups.

2¹¹:M₂₄ - containing Sylow 2-subgroups and Sylow 3-subgroups; also containing 2¹¹:(M₂₂:2), centralizer of involution of class 2B
2¹⁺¹².3.(M₂₂:2) - centralizer of involution of class 2A - containing Sylow 2-subgroups and Sylow 3-subgroups
2¹⁰:PSL(5,2)
2³⁺¹².(S₅ × PSL(3,2)) - containing Sylow 2-subgroups
U₃(11):2
M₂₂:2
11¹⁺²:(5 × GL(2,3)) - normalizer of Sylow 11-subgroup
PSL(2,32):5
PGL(2,23)
U₃(3) - containing Sylow 3-subgroups
29:28 Frobenius group
43:14 Frobenius group
37:12 Frobenius group

A Sylow 3-subgroup is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3

References

Aschbacher, Michael; Segev, Yoav (1991), "The uniqueness of groups of type J₄", Inventiones Mathematicae 105 (3): 589–607, doi:10.1007/BF01232280, ISSN 0020-9910, MR 1117152
D.J. Benson The simple group J₄, PhD Thesis, Cambridge 1981, http://www.maths.abdn.ac.uk/~bensondj/papers/b/benson/the-simple-group-J4.pdf
Ivanov, A. A. (1992), "A presentation for J₄", Proceedings of the London Mathematical Society. Third Series 64 (2): 369–396, doi:10.1112/plms/s3-64.2.369, ISSN 0024-6115, MR 1143229
Ivanov, A. A.; Meierfrankenfeld, Ulrich (1999), "A computer-free construction of J₄", Journal of Algebra 219 (1): 113–172, doi:10.1006/jabr.1999.7851, ISSN 0021-8693, MR 1707666
Ivanov, A. A. The fourth Janko group. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2004. xvi+233 pp. ISBN 0-19-852759-4 MR 2124803
Z. Janko, A new finite simple group of order 86,775,570,046,077,562,880 which possesses M₂₄ and the full covering group of M₂₂ as subgroups, J. Algebra 42 (1976) 564-596.doi:10.1016/0021-8693(76)90115-0 (The title of this paper is incorrect, as the full covering group of M₂₂ was later discovered to be larger: center of order 12, not 6.)
Kleidman, Peter B.; Wilson, Robert A. (1988), "The maximal subgroups of J₄", Proceedings of the London Mathematical Society. Third Series 56 (3): 484–510, doi:10.1112/plms/s3-56.3.484, ISSN 0024-6115, MR 931511
S. P. Norton The construction of J₄ in The Santa Cruz conference on finite groups (Ed. Cooperstein, Mason) Amer. Math. Soc 1980.

External links

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