Janko group J₄

J₄ is one of the 26 Sporadic groups. Zvonimir Janko found J₄ in 1975 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 with 2 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:A₈.

The Schur multiplier and the outer automorphism group are both trivial.

Since 37 and 43 are not supersingular primes, J₄ cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.

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.

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

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

Kleidman & Wilson (1988) found the 13 conjugacy classes of maximal subgroups of J₄ as follows:

• 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.

• 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, S2CID 121529060
• D.J. Benson The simple group J₄, PhD Thesis, Cambridge 1981, https://web.archive.org/web/20110610013308/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. (2004). The Fourth Janko Group. Oxford Mathematical Monographs. Oxford: Clarendon Press. ISBN 0-19-852759-4.MR2124803
• 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 0931511
• S. P. Norton The construction of J₄ in The Santa Cruz conference on finite groups (Ed. Cooperstein, Mason) Amer. Math. Soc 1980.

• MathWorld: Janko Groups
• Atlas of Finite Group Representations: J₄ version 2
• Atlas of Finite Group Representations: J₄ version 3